{
  "index": "1973-B-5",
  "type": "ALG",
  "tag": [
    "ALG"
  ],
  "difficulty": "",
  "question": "B-5. (a) Let \\( z \\) be a solution of the quadratic equation\n\\[\na z^{2}+b z+c=0\n\\]\nand let \\( n \\) be a positive integer. Show that \\( z \\) can be expressed as a rational function of \\( z^{n}, a, b, c \\).\n(b) Using (a) or by any other means, express \\( x \\) as a rational function of \\( x^{3} \\) and \\( x+(1 / x) \\). (Display your answer explicitly in a clearly visible form.)\n[By a rational function of several variables, we mean a quotient of polynomials in those variables, the polynomials having rational numbers as coefficients, and the denominator being not identically zero. Thus to obtain \\( x \\) as a rational function of \\( u=x^{2} \\) and \\( v= \\) \\( x+(1 / x) \\), we could write \\( x=(u+1) / v \\).",
  "solution": "B-5. (a) Let \\( r=-b / a \\) and \\( s=-c / a \\). Let polynomials \\( p_{n} \\) and \\( q_{n} \\) in \\( r \\) and \\( s \\) be defined by the initial conditions \\( p_{0}=0, p_{1}=1, q_{0}=1 \\), and \\( q_{1}=0 \\) and the recursion formulas \\( p_{n}=r p_{n-1}+s p_{n-2} \\) and \\( q_{n}=r q_{n-1}+s q_{n-2} \\) for \\( n>1 \\). Using \\( z^{n}=r z^{n-1}+s z^{n-2} \\) and mathematical induction, one proves that \\( z^{n}=p_{n} z+q_{n} \\) and that all the coefficients in \\( p_{n}(r, s) \\) are positive. Then multiplying numerator and denominator of the right hand side of \\( z=\\left[z^{n}-q_{n}(-b / a,-c / a)\\right] / p_{n}(-b / a,-c / a) \\) by the proper power of \\( a \\) leads to \\( z=F\\left(z^{n}, a, b, c\\right) / G(a, b, c) \\), where \\( F \\) and \\( G \\) are polynomials with integer coefficients. Since all the coefficients in \\( p_{n}(r, s) \\) are positive, the same is true of \\( G(a, b, c) \\). Therefore \\( G(a, b, c) \\) is not identically zero and \\( F / G \\) is the desired rational function.\n(b) Let \\( v=x+(1 / x) \\). Then \\( x^{2}-v x+1=0 \\). Using (a) with \\( z \\) replaced by \\( x \\), one finds that \\( x^{3}=p_{3} x+q_{3} \\) with \\( p_{3}=v^{2}-1 \\) and \\( q_{3}=-v \\). Then\n\\[\nx=\\left(x^{3}-q_{3}\\right) / p_{3}=\\left(x^{3}+v\\right) /\\left(v^{2}-1\\right) .\n\\]",
  "vars": [
    "z",
    "x",
    "n",
    "u",
    "v"
  ],
  "params": [
    "a",
    "b",
    "c",
    "r",
    "s",
    "p_0",
    "p_1",
    "p_n",
    "p_n-1",
    "p_n-2",
    "q_0",
    "q_1",
    "q_n",
    "q_n-1",
    "q_n-2",
    "F",
    "G"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "z": "rootvariable",
        "x": "basevariable",
        "n": "positiveindex",
        "u": "squarevalue",
        "v": "suminverse",
        "a": "coeffalpha",
        "b": "coeffbeta",
        "c": "coeffgamma",
        "r": "coeffrho",
        "s": "coeffsigma",
        "p_0": "polyzero",
        "p_1": "polyone",
        "p_n": "polyindex",
        "p_n-1": "polypreindex",
        "p_n-2": "polypreprev",
        "q_0": "quasizero",
        "q_1": "quasione",
        "q_n": "quasiindex",
        "q_n-1": "quasipreindex",
        "q_n-2": "quasipreprev",
        "F": "numerpoly",
        "G": "denompoly"
      },
      "question": "B-5. (a) Let \\( rootvariable \\) be a solution of the quadratic equation\n\\[\ncoeffalpha rootvariable^{2}+coeffbeta rootvariable+coeffgamma=0\n\\]\nand let \\( positiveindex \\) be a positive integer. Show that \\( rootvariable \\) can be expressed as a rational function of \\( rootvariable^{positiveindex}, coeffalpha, coeffbeta, coeffgamma \\).\n(b) Using (a) or by any other means, express \\( basevariable \\) as a rational function of \\( basevariable^{3} \\) and \\( basevariable+(1 / basevariable) \\). (Display your answer explicitly in a clearly visible form.)\n[By a rational function of several variables, we mean a quotient of polynomials in those variables, the polynomials having rational numbers as coefficients, and the denominator being not identically zero. Thus to obtain \\( basevariable \\) as a rational function of \\( squarevalue=basevariable^{2} \\) and \\( suminverse= \\) \\( basevariable+(1 / basevariable) \\), we could write \\( basevariable=(squarevalue+1) / suminverse \\).",
      "solution": "B-5. (a) Let \\( coeffrho=-coeffbeta / coeffalpha \\) and \\( coeffsigma=-coeffgamma / coeffalpha \\). Let polynomials \\( polyindex \\) and \\( quasiindex \\) in \\( coeffrho \\) and \\( coeffsigma \\) be defined by the initial conditions \\( polyzero=0, polyone=1, quasizero=1 \\), and \\( quasione=0 \\) and the recursion formulas \\( polyindex=coeffrho polypreindex+coeffsigma polypreprev \\) and \\( quasiindex=coeffrho quasipreindex+coeffsigma quasipreprev \\) for \\( positiveindex>1 \\). Using \\( rootvariable^{positiveindex}=coeffrho rootvariable^{positiveindex-1}+coeffsigma rootvariable^{positiveindex-2} \\) and mathematical induction, one proves that \\( rootvariable^{positiveindex}=polyindex rootvariable+quasiindex \\) and that all the coefficients in \\( polyindex(coeffrho, coeffsigma) \\) are positive. Then multiplying numerator and denominator of the right hand side of \\( rootvariable=\\left[rootvariable^{positiveindex}-quasiindex(-coeffbeta / coeffalpha,-coeffgamma / coeffalpha)\\right] / polyindex(-coeffbeta / coeffalpha,-coeffgamma / coeffalpha) \\) by the proper power of \\( coeffalpha \\) leads to \\( rootvariable=numerpoly\\left(rootvariable^{positiveindex}, coeffalpha, coeffbeta, coeffgamma\\right) / denompoly(coeffalpha, coeffbeta, coeffgamma) \\), where numerpoly and denompoly are polynomials with integer coefficients. Since all the coefficients in \\( polyindex(coeffrho, coeffsigma) \\) are positive, the same is true of \\( denompoly(coeffalpha, coeffbeta, coeffgamma) \\). Therefore \\( denompoly(coeffalpha, coeffbeta, coeffgamma) \\) is not identically zero and \\( numerpoly / denompoly \\) is the desired rational function.\n(b) Let \\( suminverse=basevariable+(1 / basevariable) \\). Then \\( basevariable^{2}-suminverse basevariable+1=0 \\). Using (a) with \\( rootvariable \\) replaced by \\( basevariable \\), one finds that \\( basevariable^{3}=p_{3} basevariable+q_{3} \\) with \\( p_{3}=suminverse^{2}-1 \\) and \\( q_{3}=-suminverse \\). Then\n\\[\nbasevariable=\\left(basevariable^{3}+suminverse\\right) /\\left(suminverse^{2}-1\\right) .\n\\]"
    },
    "descriptive_long_confusing": {
      "map": {
        "z": "pineapple",
        "x": "hamburger",
        "n": "carousel",
        "u": "telescope",
        "v": "marigold",
        "a": "backpack",
        "b": "suitcase",
        "c": "zeppelin",
        "r": "dinosaur",
        "s": "chocolate",
        "p_0": "lighthouse",
        "p_1": "snowflake",
        "p_n": "watermelon",
        "p_n-1": "butterfly",
        "p_n-2": "landscape",
        "q_0": "guitarist",
        "q_1": "astronaut",
        "q_n": "candlelight",
        "q_n-1": "helicopter",
        "q_n-2": "pomegranate",
        "F": "barnstorm",
        "G": "horseshoe"
      },
      "question": "B-5. (a) Let \\( pineapple \\) be a solution of the quadratic equation\n\\[\nbackpack\\, pineapple^{2}+suitcase\\, pineapple+zeppelin=0\n\\]\nand let \\( carousel \\) be a positive integer. Show that \\( pineapple \\) can be expressed as a rational function of \\( pineapple^{carousel}, backpack, suitcase, zeppelin \\).\n(b) Using (a) or by any other means, express \\( hamburger \\) as a rational function of \\( hamburger^{3} \\) and \\( hamburger+(1 / hamburger) \\). (Display your answer explicitly in a clearly visible form.)\n[By a rational function of several variables, we mean a quotient of polynomials in those variables, the polynomials having rational numbers as coefficients, and the denominator being not identically zero. Thus to obtain \\( hamburger \\) as a rational function of \\( telescope=hamburger^{2} \\) and \\( marigold= \\) \\( hamburger+(1 / hamburger) \\), we could write \\( hamburger=(telescope+1) / marigold \\).",
      "solution": "B-5. (a) Let \\( dinosaur=-suitcase / backpack \\) and \\( chocolate=-zeppelin / backpack \\). Let polynomials \\( watermelon \\) and \\( candlelight \\) in \\( dinosaur \\) and \\( chocolate \\) be defined by the initial conditions \\( lighthouse=0, snowflake=1, guitarist=1 \\), and \\( astronaut=0 \\) and the recursion formulas \\( watermelon=dinosaur butterfly+chocolate landscape \\) and \\( candlelight=dinosaur helicopter+chocolate pomegranate \\) for \\( carousel>1 \\). Using \\( pineapple^{carousel}=dinosaur pineapple^{carousel-1}+chocolate pineapple^{carousel-2} \\) and mathematical induction, one proves that \\( pineapple^{carousel}=watermelon pineapple+candlelight \\) and that all the coefficients in \\( watermelon(dinosaur, chocolate) \\) are positive. Then multiplying numerator and denominator of the right hand side of \\( pineapple=\\left[pineapple^{carousel}-candlelight(-suitcase / backpack,-zeppelin / backpack)\\right] / watermelon(-suitcase / backpack,-zeppelin / backpack) \\) by the proper power of \\( backpack \\) leads to \\( pineapple=barnstorm\\left(pineapple^{carousel}, backpack, suitcase, zeppelin\\right) / horseshoe(backpack, suitcase, zeppelin) \\), where \\( barnstorm \\) and \\( horseshoe \\) are polynomials with integer coefficients. Since all the coefficients in \\( watermelon(dinosaur, chocolate) \\) are positive, the same is true of \\( horseshoe(backpack, suitcase, zeppelin) \\). Therefore \\( horseshoe(backpack, suitcase, zeppelin) \\) is not identically zero and \\( barnstorm / horseshoe \\) is the desired rational function.\n(b) Let \\( marigold=hamburger+(1 / hamburger) \\). Then \\( hamburger^{2}-marigold\\, hamburger+1=0 \\). Using (a) with \\( pineapple \\) replaced by \\( hamburger \\), one finds that \\( hamburger^{3}=p_{3} hamburger+q_{3} \\) with \\( p_{3}=marigold^{2}-1 \\) and \\( q_{3}=-marigold \\). Then\n\\[\nhamburger=\\left(hamburger^{3}-q_{3}\\right) / p_{3}=\\left(hamburger^{3}+marigold\\right) /\\left(marigold^{2}-1\\right) .\n\\]"
    },
    "descriptive_long_misleading": {
      "map": {
        "z": "summitpoint",
        "x": "knownvar",
        "n": "negative",
        "u": "rootvalue",
        "v": "difference",
        "a": "trailing",
        "b": "outercoef",
        "c": "firstcoef",
        "r": "productor",
        "s": "collective",
        "p_0": "finalpoly",
        "p_1": "initialpoly",
        "p_n": "genericpoly",
        "p_n-1": "previouspoly",
        "p_n-2": "preprevpoly",
        "q_0": "finalcoef",
        "q_1": "initialcoef",
        "q_n": "genericcoef",
        "q_n-1": "previouscoef",
        "q_n-2": "preprevcoef",
        "F": "denominator",
        "G": "numerator"
      },
      "question": "B-5. (a) Let \\( summitpoint \\) be a solution of the quadratic equation\n\\[\ntrailing \\, summitpoint^{2}+outercoef \\, summitpoint+firstcoef=0\n\\]\nand let \\( negative \\) be a positive integer. Show that \\( summitpoint \\) can be expressed as a rational function of \\( summitpoint^{negative}, trailing, outercoef, firstcoef \\).\n(b) Using (a) or by any other means, express \\( knownvar \\) as a rational function of \\( knownvar^{3} \\) and \\( knownvar+(1 / knownvar) \\). (Display your answer explicitly in a clearly visible form.)\n[By a rational function of several variables, we mean a quotient of polynomials in those variables, the polynomials having rational numbers as coefficients, and the denominator being not identically zero. Thus to obtain \\( knownvar \\) as a rational function of \\( rootvalue=knownvar^{2} \\) and \\( difference= \\) \\( knownvar+(1 / knownvar) \\), we could write \\( knownvar=(rootvalue+1) / difference \\).",
      "solution": "B-5. (a) Let \\( productor=-outercoef / trailing \\) and \\( collective=-firstcoef / trailing \\). Let polynomials \\( genericpoly \\) and \\( genericcoef \\) in \\( productor \\) and \\( collective \\) be defined by the initial conditions \\( finalpoly=0, initialpoly=1, finalcoef=1 \\), and \\( initialcoef=0 \\) and the recursion formulas \\( genericpoly=productor previouspoly+collective preprevpoly \\) and \\( genericcoef=productor previouscoef+collective preprevcoef \\) for \\( negative>1 \\). Using \\( summitpoint^{negative}=productor summitpoint^{negative-1}+collective summitpoint^{negative-2} \\) and mathematical induction, one proves that \\( summitpoint^{negative}=genericpoly summitpoint+genericcoef \\) and that all the coefficients in \\( genericpoly(productor, collective) \\) are positive. Then multiplying numerator and denominator of the right hand side of \\( summitpoint=\\left[summitpoint^{negative}-genericcoef(-outercoef / trailing,-firstcoef / trailing)\\right] / genericpoly(-outercoef / trailing,-firstcoef / trailing) \\) by the proper power of \\( trailing \\) leads to \\( summitpoint=denominator\\left(summitpoint^{negative}, trailing, outercoef, firstcoef\\right) / numerator(trailing, outercoef, firstcoef) \\), where \\( denominator \\) and \\( numerator \\) are polynomials with integer coefficients. Since all the coefficients in \\( genericpoly(productor, collective) \\) are positive, the same is true of \\( numerator(trailing, outercoef, firstcoef) \\). Therefore \\( numerator(trailing, outercoef, firstcoef) \\) is not identically zero and \\( denominator / numerator \\) is the desired rational function.\n(b) Let \\( difference=knownvar+(1 / knownvar) \\). Then \\( knownvar^{2}-difference knownvar+1=0 \\). Using (a) with \\( summitpoint \\) replaced by \\( knownvar \\), one finds that \\( knownvar^{3}=genericpoly knownvar+genericcoef \\) with \\( genericpoly=difference^{2}-1 \\) and \\( genericcoef=-difference \\). Then\n\\[\nknownvar=\\left(knownvar^{3}-genericcoef\\right) / genericpoly=\\left(knownvar^{3}+difference\\right) /\\left(difference^{2}-1\\right) .\n\\]"
    },
    "garbled_string": {
      "map": {
        "z": "qzxwvtnp",
        "x": "hjgrksla",
        "n": "vfglimqu",
        "u": "tbskdnea",
        "v": "pxjrmhao",
        "a": "kfwlanop",
        "b": "cszmjdqe",
        "c": "ylgtbvio",
        "r": "hqzfldwa",
        "s": "wepqmxor",
        "p_0": "lidwqnez",
        "p_1": "gpyfkrso",
        "p_n": "jxrhabks",
        "p_n-1": "mcltvezp",
        "p_n-2": "dwrqkgvz",
        "q_0": "fakmesot",
        "q_1": "ziqbrduy",
        "q_n": "twlmpsha",
        "q_n-1": "bsgwxyif",
        "q_n-2": "uhvozrpa",
        "F": "neuxgkdc",
        "G": "rovjtdal"
      },
      "question": "B-5. (a) Let \\( qzxwvtnp \\) be a solution of the quadratic equation\n\\[\nkfwlanop qzxwvtnp^{2}+cszmjdqe qzxwvtnp+ylgtbvio=0\n\\]\nand let \\( vfglimqu \\) be a positive integer. Show that \\( qzxwvtnp \\) can be expressed as a rational function of \\( qzxwvtnp^{vfglimqu}, kfwlanop, cszmjdqe, ylgtbvio \\).\n(b) Using (a) or by any other means, express \\( hjgrksla \\) as a rational function of \\( hjgrksla^{3} \\) and \\( hjgrksla+(1 / hjgrksla) \\). (Display your answer explicitly in a clearly visible form.)\n[By a rational function of several variables, we mean a quotient of polynomials in those variables, the polynomials having rational numbers as coefficients, and the denominator being not identically zero. Thus to obtain \\( hjgrksla \\) as a rational function of \\( tbskdnea=hjgrksla^{2} \\) and \\( pxjrmhao= hjgrksla+(1 / hjgrksla) \\), we could write \\( hjgrksla=(tbskdnea+1) / pxjrmhao \\).",
      "solution": "B-5. (a) Let \\( hqzfldwa=-cszmjdqe / kfwlanop \\) and \\( wepqmxor=-ylgtbvio / kfwlanop \\). Let polynomials \\( jxrhabks \\) and \\( twlmpsha \\) in \\( hqzfldwa \\) and \\( wepqmxor \\) be defined by the initial conditions \\( lidwqnez=0, gpyfkrso=1, fakmesot=1 \\), and \\( ziqbrduy=0 \\) and the recursion formulas \\( jxrhabks=hqzfldwa\\,mcltvezp+wepqmxor\\,dwrqkgvz \\) and \\( twlmpsha=hqzfldwa\\,bsgwxyif+wepqmxor\\,uhvozrpa \\) for \\( vfglimqu>1 \\). Using \\( qzxwvtnp^{vfglimqu}=hqzfldwa\\,qzxwvtnp^{vfglimqu-1}+wepqmxor\\,qzxwvtnp^{vfglimqu-2} \\) and mathematical induction, one proves that \\( qzxwvtnp^{vfglimqu}=jxrhabks\\,qzxwvtnp+twlmpsha \\) and that all the coefficients in \\( jxrhabks(hqzfldwa, wepqmxor) \\) are positive. Then multiplying numerator and denominator of the right hand side of \n\\[\nqzxwvtnp=\\left[qzxwvtnp^{vfglimqu}-twlmpsha(-cszmjdqe / kfwlanop,-ylgtbvio / kfwlanop)\\right]\\big/ jxrhabks(-cszmjdqe / kfwlanop,-ylgtbvio / kfwlanop)\n\\]\nby the proper power of \\( kfwlanop \\) leads to \n\\[\nqzxwvtnp=neuxgkdc\\left(qzxwvtnp^{vfglimqu}, kfwlanop, cszmjdqe, ylgtbvio\\right) / rovjtdal(kfwlanop, cszmjdqe, ylgtbvio),\n\\]\nwhere \\( neuxgkdc \\) and \\( rovjtdal \\) are polynomials with integer coefficients. Since all the coefficients in \\( jxrhabks(hqzfldwa, wepqmxor) \\) are positive, the same is true of \\( rovjtdal(kfwlanop, cszmjdqe, ylgtbvio) \\). Therefore \\( rovjtdal(kfwlanop, cszmjdqe, ylgtbvio) \\) is not identically zero and \\( neuxgkdc / rovjtdal \\) is the desired rational function.\n(b) Let \\( pxjrmhao=hjgrksla+(1 / hjgrksla) \\). Then \\( hjgrksla^{2}-pxjrmhao\\,hjgrksla+1=0 \\). Using (a) with \\( qzxwvtnp \\) replaced by \\( hjgrksla \\), one finds that \\( hjgrksla^{3}=p_{3}\\,hjgrksla+q_{3} \\) with \\( p_{3}=pxjrmhao^{2}-1 \\) and \\( q_{3}=-pxjrmhao \\). Then\n\\[\nhjgrksla=\\left(hjgrksla^{3}-q_{3}\\right) / p_{3}=\\left(hjgrksla^{3}+pxjrmhao\\right) /\\left(pxjrmhao^{2}-1\\right) .\n\\]"
    },
    "kernel_variant": {
      "question": "(a)   Let  \n\\[\na z^{3}+b z^{2}+c z+d=0\\qquad\\qquad (1)\n\\]  \nbe a cubic with rational coefficients and assume  \n\\[\na\\neq 0,\\qquad d\\neq 0 ,\\qquad   \n\\Delta:=b^{2}c^{2}-4ac^{3}-4b^{3}d-27a^{2}d^{2}+18a b c d\\neq 0 .\n\\]  \nIntroduce the reduced parameters  \n\\[\np=-\\dfrac{b}{a},\\qquad q=-\\dfrac{c}{a},\\qquad r=-\\dfrac{d}{a}. \\tag{2}\n\\]\n\nFix an integer $n\\ge 2$.\n\n(i)  Show that there exist unique polynomials  \n\\[\nP_{n}(p,q,r),\\;B_{n}(p,q,r),\\;C_{n}(p,q,r)\\in\\mathbb{Q}[p,q,r]\\tag{3}\n\\]  \nsuch that  \n\\[\nz^{n}=P_{n}+B_{n}z+C_{n}z^{2}. \\tag{4}\n\\]\n\n(ii)  Put  \n\\[\nK_{n}:=\\mathbb{Q}(a,b,c,d,z^{n})\n\\]\nand consider over $K_{n}$ the quadratic polynomial\n\\[\nQ_{n}(T)=C_{n}\\,T^{2}+B_{n}\\,T+\\bigl(P_{n}-z^{n}\\bigr). \\tag{5}\n\\]\n\n1.  Prove that $Q_{n}(z)=0$ and deduce\n\n   •  if $(B_{n},C_{n})=(0,0)$ then $z^{n}\\in\\mathbb{Q}(a,b,c,d)$; hence  \n     $K_{n}=\\mathbb{Q}(a,b,c,d)$ and  \n     \\[\n       [K_{n}(z):K_{n}]=\\deg_{\\mathbb{Q}}(z)\\in\\{1,2,3\\},\n     \\]  \n     the value $3$ occurring precisely when the cubic $(1)$ is\n     irreducible over $\\mathbb{Q}$;\n\n   •  if $(B_{n},C_{n})\\neq(0,0)$ then $[K_{n}(z):K_{n}]\\le 2$.\n\n   Treat separately the sub–cases  \n\n   – $C_{n}=0,\\;B_{n}\\neq 0$ (linear case),  \n\n   – $C_{n}\\neq 0$ (proper quadratic).\n\n   Show moreover that for an irreducible cubic $(1)$ one has  \n   \\[\n   (B_{n},C_{n})=(0,0)\n   \\;\\Longleftrightarrow\\;\n   p=q=0\\text{ and }3\\mid n. \\tag{6}\n   \\]\n\n2.  Assuming $C_{n}\\neq 0$, derive the quadratic–formula identity  \n\\[\nz=\\dfrac{-B_{n}\\pm\\sqrt{\\Delta_{n}}}{2C_{n}},\\qquad \n\\Delta_{n}=B_{n}^{2}+4C_{n}\\bigl(z^{n}-P_{n}\\bigr)\\in K_{n}. \\tag{7}\n\\]\n\n3.  Assume now that  \n\n   •  the cubic $(1)$ is irreducible over $\\mathbb{Q}$;  \n   •  its discriminant $\\Delta$ is not a square in $\\mathbb{Q}$.\n\n   Let $z_{1}=z,z_{2},z_{3}$ be the roots of $(1)$ and define  \n   \\[\n   \\mu:=\\dfrac{z_{2}}{z_{1}},\\qquad \n   \\nu:=\\dfrac{z_{3}}{z_{1}},\\qquad \n   R:=\\{1,\\mu,\\nu,\\mu^{-1},\\nu^{-1},\\mu/\\nu\\}. \\tag{8}\n   \\]\n\n   (a)  Prove that  \n        \\[\n        [\\mathbb{Q}(\\mu):\\mathbb{Q}]\n        =[\\,\\mathbb{Q}(\\nu):\\mathbb{Q}\\,]\\in\\{2,3,6\\}.\n        \\]\n        Show further that if one of $\\mu,\\nu$ is a root of unity, then its\n        order lies in $\\{1,2,3,4,6\\}$.\n\n   (b)  Put $L$ for the splitting field of $(1)$ and\n        \\[\n          H_{n}:=\\{\\tau\\in\\operatorname{Gal}(L/\\mathbb{Q})\\mid\n          (\\tau(z)/z)^{\\,n}=1\\}.\n        \\]\n        Establish the chain of equivalences  \n        \\[\n        \\bigl[(B_{n},C_{n})\\neq(0,0)\\bigr]\\;\\wedge\\;\n        \\Bigl(Q_{n}\\text{ irreducible over }K_{n}\\Bigr)\n        \\;\\Longleftrightarrow\\;\n        z\\notin K_{n}\n        \\;\\Longleftrightarrow\\;\n        R\\setminus\\{1\\}\\text{ contains an }n\\text{-th root of unity}. \\tag{9}\n        \\]\n\n   (c)  Deduce:\n\n        •  If no element of $R\\setminus\\{1\\}$ is a root of unity, then\n          $(B_{n},C_{n})\\neq(0,0)$ and $Q_{n}(T)$ splits over $K_{n}$,\n          so $z\\in K_{n}$ for every $n\\ge 2$.\n\n        •  If the subgroup $\\langle\\mu,\\nu\\rangle\\subset L^{\\times}$ is\n          finite of order $m\\in\\{2,3,4,6\\}$, then, for all $n\\ge 2$,\n          \\[\n          Q_{n}(T)\\text{ is irreducible over }K_{n}\n          \\;\\Longleftrightarrow\\;\n          \\gcd(m,n)\\neq 1,\n          \\]\n          whereas $Q_{n}(T)$ splits (and therefore $z\\in K_{n}$)\n          when $\\gcd(m,n)=1$.\n\n(b)  For every non–zero complex number $x$ put  \n\\[\nu=x+\\dfrac{1}{x}.\n\\]\nProve the inversion identity  \n\\[\nx=\\dfrac{x^{7}+u^{5}-4u^{3}+3u}{\\,u^{6}-5u^{4}+6u^{2}-1\\,} \\tag{10}\n\\]\nand verify that the denominator is not the zero polynomial, so the\nformula is valid for all $x$ satisfying\n$u^{6}-5u^{4}+6u^{2}-1\\neq 0$.\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "solution": "(a)\n\n(i)  \\emph{Existence and uniqueness of the reduction polynomials.}  \nDividing $(1)$ by $a$ gives the monic relation  \n\\[\nz^{3}=p z^{2}+q z+r. \\tag{11}\n\\]\nPut  \n\\[\n\\mathcal{V}:=\\operatorname{Span}_{\\mathbb{Q}}\\{\\,1,z,z^{2}\\,\\}.\n\\]\nBecause any power $z^{k}\\;(k\\ge 3)$ can be reduced with $(11)$ to an\nelement of $\\mathcal{V}$, there exist polynomials\n$P_{k},B_{k},C_{k}\\in\\mathbb{Q}[p,q,r]$ such that\n\\[\nz^{k}=P_{k}+B_{k}z+C_{k}z^{2}\\qquad(k\\ge 0). \\tag{12}\n\\]\n\nWork in the fraction field $\\mathbb{Q}(p,q,r)$.\nRelation $(11)$ is the \\emph{generic} monic cubic and is irreducible\nover this field; consequently $1,z,z^{2}$ are linearly independent\nover $\\mathbb{Q}(p,q,r)$.\nHence representation $(12)$ is unique, so the reduction polynomials in\n$(4)$ are well defined.\n\nA convenient way to compute them is the matrix recurrence  \n\\[\n\\mathbf{v}_{k}:=\\begin{pmatrix}P_{k}\\\\[2pt]B_{k}\\\\[2pt]C_{k}\\end{pmatrix},\n\\qquad\n\\mathbf{v}_{k+3}=p\\,\\mathbf{v}_{k+2}+q\\,\\mathbf{v}_{k+1}+r\\,\\mathbf{v}_{k},\n\\tag{13}\n\\]\nstarted from  \n$\\mathbf{v}_{0}=(1,0,0)^{\\mathrm T},\\;\n  \\mathbf{v}_{1}=(0,1,0)^{\\mathrm T},\\;\n  \\mathbf{v}_{2}=(0,0,1)^{\\mathrm T}$.\n\nA special phenomenon occurs when $p=q=0$: then $z^{3}=r$ and one\nchecks immediately from $(13)$ that  \n\\[\n(B_{n},C_{n})=(0,0)\\quad\\Longleftrightarrow\\quad 3\\mid n. \\tag{14}\n\\]\n\n%---------------------------------------------------------------------------\n(ii-1)  Substituting $T=z$ in $Q_{n}(T)$ gives $Q_{n}(z)=0$.\n\n\\emph{Case $(B_{n},C_{n})=(0,0)$.}  \nThen $z^{n}=P_{n}\\in\\mathbb{Q}(a,b,c,d)$, so\n$K_{n}=\\mathbb{Q}(a,b,c,d)$ and consequently  \n\\[\n[K_{n}(z):K_{n}]=\\deg_{\\mathbb{Q}}(z)\\in\\{1,2,3\\},\n\\]\nthe value $3$ occurring exactly when the cubic $(1)$ is irreducible\nover $\\mathbb{Q}$.\n\n\\emph{Case $(B_{n},C_{n})\\neq(0,0)$.}\n\n• If $C_{n}=0$ but $B_{n}\\neq 0$, then  \n\\[\nz=\\dfrac{z^{n}-P_{n}}{B_{n}}\\in K_{n},\n\\]\nso $[K_{n}(z):K_{n}]=1\\le 2$.\n\n• If $C_{n}\\neq 0$, then $Q_{n}(T)$ is a genuine quadratic\nwith root $z$; hence $[K_{n}(z):K_{n}]\\le 2$.\n\n%---------------------------------------------------------------------------\n\\textbf{The equivalence $(6)$ for irreducible cubics – corrected proof.}\n\nAssume from now on that the cubic $(1)$ is \\emph{irreducible} over\n$\\mathbb{Q}$.\nLet $L$ be its splitting field.  \nBecause $\\Delta\\notin\\mathbb{Q}^{2}$, one has  \n\\[\n\\operatorname{Gal}(L/\\mathbb{Q})\\;\\cong\\;S_{3}. \\tag{15}\n\\]\nChoose the $3$-cycle  \n\\[\n\\sigma:\\; z_{1}\\mapsto z_{2},\\; z_{2}\\mapsto z_{3},\\; z_{3}\\mapsto z_{1},\n\\qquad z_{1}=z. \\tag{16}\n\\]\nPut  \n\\[\n\\mu:=\\frac{z_{2}}{z_{1}},\\qquad\n\\nu:=\\frac{z_{3}}{z_{1}}. \\tag{17}\n\\]\n\n\\subsubsection*{Step A:  $(B_{n},C_{n})=(0,0)\\;\\Longrightarrow\\;\\mu^{\\,n}=1$.}\n\nIndeed, $(B_{n},C_{n})=(0,0)$ means $z^{n}\\in\\mathbb{Q}$.  \nApplying $\\sigma$ we obtain\n\\[\nz^{\\,n}=\\sigma\\bigl(z^{\\,n}\\bigr)=z_{2}^{\\,n}\n        =(\\mu z)^{\\,n}=\\mu^{\\,n}z^{\\,n},\n\\]\nhence $\\mu^{\\,n}=1$.\nAn identical argument with $\\sigma^{2}$ gives $\\nu^{\\,n}=1$.\n\n\\subsubsection*{Step B:  If two distinct roots differ by a root of unity,\nthen that root of unity is \\emph{necessarily} a primitive third root and\n$p=q=0$.}\n\n\\begin{lemma}\\label{lem:new}\nLet $f(X)=X^{3}+bX^{2}+cX+d\\in\\mathbb{Q}[X]$ be irreducible and let\n$z_{1},z_{2}$ be two distinct roots with\n$z_{2}=\\zeta\\,z_{1}$ for some root of unity $\\zeta\\neq 1$.\nThen $\\zeta^{3}=1$ and $b=c=0$.\n\\end{lemma}\n\n\\emph{Proof.}\nDefine\n\\[\ng(X):=f(X)-f(\\zeta X)\n          =(1-\\zeta^{3})X^{3}+b(1-\\zeta^{2})X^{2}+c(1-\\zeta)X. \\tag{18}\n\\]\nBecause $f(z_{1})=f(z_{2})=0$, the number $z_{1}$ is a common root of\n$f$ and $g$.\n\n$\\bullet$ If $\\zeta^{3}\\neq 1$, then the coefficient $1-\\zeta^{3}$ is\nnon–zero, so $\\deg g=3$ and $g$ is \\emph{not} a scalar multiple of $f$\n(their coefficients of $X^{3}$ differ).\nHence $\\gcd(f,g)$ is a non-constant proper divisor of $f$, contradicting\nthe irreducibility of $f$.\nTherefore we must have $\\zeta^{3}=1$.\n\n$\\bullet$ Put now $\\zeta^{3}=1$ with $\\zeta\\neq 1$ (so $\\zeta$ is a\nprimitive third root of unity).\nThen $g(X)=b(1-\\zeta^{2})X^{2}+c(1-\\zeta)X$ has degree at most $2$ and\nstill vanishes at $z_{1}$.\nIf at least one of $b$ or $c$ is non-zero, the degree of $g$ is\n$1$ or $2$ and $\\gcd(f,g)$ is again a proper non–constant factor of\n$f$, impossible.\nConsequently $b=c=0$, i.e.\\ $p=q=0$. \\qed\n\n\\smallskip\nLemma \\ref{lem:new} shows that if $\\mu$ (or $\\nu$) is a root of unity,\nthen $\\mu^{3}=1$ and $p=q=0$.\n\n\\subsubsection*{Step C:  Completion of $(6)$.}\n\nAssume $(B_{n},C_{n})=(0,0)$.  \nStep A gives $\\mu^{\\,n}=1$; if $\\mu\\neq 1$, Lemma \\ref{lem:new} yields\n$\\mu^{3}=1$ and $p=q=0$.  \nBecause $\\mu^{\\,3}=1$, necessarily $3\\mid n$.\nIf $\\mu=1$ then $z_{1}=z_{2}$, impossible; hence $\\mu\\neq 1$ and the\nprevious conclusion applies.\n\nConversely, if $p=q=0$ then $z^{3}=r$ and relation $(14)$ shows that\n$(B_{n},C_{n})=(0,0)$ exactly when $3\\mid n$.\nThus\n\\[\n(B_{n},C_{n})=(0,0)\n\\;\\Longleftrightarrow\\;\np=q=0\\text{ and }3\\mid n,\n\\]\nestablishing $(6)$.\n\n%---------------------------------------------------------------------------\n(ii-2)  When $C_{n}\\neq 0$, the quadratic formula applied to\n$Q_{n}(T)=0$ yields $(7)$.\n\n%---------------------------------------------------------------------------\n(ii-3)  Assume now $(1)$ irreducible and $\\Delta\\notin\\mathbb{Q}^{2}$.\nWe work with the notations in $(8)$.\n\n(α)  The stabiliser of $\\mu$ (or $\\nu$) in $S_{3}$ can have order\n$1,2$ or $3$; its index is therefore $6,3$ or $2$, giving\n$[\\mathbb{Q}(\\mu):\\mathbb{Q}]\\in\\{6,3,2\\}$.\nA root of unity of degree at most $6$ has order in\n$\\{1,2,3,4,6\\}$, establishing the first assertions.\n\n(β)  For $\\tau\\in\\operatorname{Gal}(L/\\mathbb{Q})$ set  \n\\[\n\\rho(\\tau):=\\dfrac{\\tau(z)}{z}\\in R .  \\tag{19}\n\\]\nBecause $\\tau(z^{\\,n})=\\rho(\\tau)^{\\,n}z^{\\,n}$,\n\\[\nH_{n}=\\bigl\\{\\tau\\in\\operatorname{Gal}(L/\\mathbb{Q})\\mid\n        \\rho(\\tau)^{\\,n}=1\\bigr\\}. \\tag{20}\n\\]\n\n(γ)  If $(B_{n},C_{n})\\neq(0,0)$,\nthen $Q_{n}$ splits over $K_{n}$ precisely when $z\\in K_{n}$,\nand is irreducible otherwise.\nUsing $(20)$ one obtains\n\\[\nQ_{n}(T)\\text{ irreducible over }K_{n}\n\\;\\Longleftrightarrow\\;\n\\exists\\tau\\in S_{3}:\\rho(\\tau)\\neq 1,\\; \\rho(\\tau)^{\\,n}=1\n\\;\\Longleftrightarrow\\;\nR\\setminus\\{1\\}\\text{ contains an }n\\text{-th root of unity}, \\tag{21}\n\\]\nwhich is the second equivalence in $(9)$.\nTogether with the cases distinguished at the beginning of (ii-1), this\ngives the whole chain $(9)$.\n\n(δ)  The assertions in part (c) follow immediately from $(21)$.\n\n%---------------------------------------------------------------------------\n(b)  \n\nFrom $u=x+\\dfrac{1}{x}$ one has  \n\\[\nx^{2}=u\\,x-1. \\tag{22}\n\\]\nRepeatedly multiply by $x$ and reduce with $(22)$:\n\\[\n\\begin{aligned}\nx^{3}&=(u^{2}-1)x-u,\\\\\nx^{4}&=(u^{3}-2u)x-(u^{2}-1),\\\\\nx^{5}&=(u^{4}-3u^{2}+1)x-u(u^{2}-2),\\\\\nx^{6}&=(u^{5}-4u^{3}+3u)x-(u^{4}-3u^{2}+1),\\\\\nx^{7}&=(u^{6}-5u^{4}+6u^{2}-1)x-(u^{5}-4u^{3}+3u). \\tag{23}\n\\end{aligned}\n\\]\nSolving the last line of $(23)$ for $x$ gives exactly the inversion\nidentity $(10)$.\nThe denominator $u^{6}-5u^{4}+6u^{2}-1$ is a non–zero polynomial, so\nthe identity is valid whenever this denominator does not vanish.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.611591",
        "was_fixed": false,
        "difficulty_analysis": "•  The original problem dealt with a quadratic and used a single power zⁿ.\n   Here we move to a *cubic*, which raises the ambient field degree from 2\n   to 3 and forces one to manipulate *three* coefficient–dependent sequences\n   simultaneously instead of one.  \n•  Part (a) demands an explicit construction of Pₙ,Bₙ,Cₙ, the derivation of a\n   pair of coupled linear equations, and the use of Cramer’s rule; none of\n   these is needed for the quadratic case.  The proof also has to guarantee\n   that some small n works for *all* cubics, invoking a non–trivial argument\n   on the coefficients.  \n•  Part (b) asks for the recovery of x from (x⁵,u).  The exponent 5 forces\n   three rounds of substitutions (compare the single round required when the\n   given power is x⁴ in the current kernel variant), and the resulting\n   polynomials are of much higher degree (degree 4 in u).  Careful algebra is\n   indispensable to avoid sign and coefficient errors.  \n•  Together, the two parts blend linear–recurrence theory, determinant\n   techniques, and clever algebraic manipulation, going noticeably beyond the\n   elementary pattern-matching that suffices for the quadratic prototype."
      }
    },
    "original_kernel_variant": {
      "question": "(a)  Let  \n\\[\na z^{3}+b z^{2}+c z+d=0 \\qquad\\qquad (1)\n\\]  \nbe a cubic with rational coefficients and assume  \n\\[\na d\\neq 0 ,\\qquad   \n\\Delta:=b^{2}c^{2}-4ac^{3}-4b^{3}d-27a^{2}d^{2}+18a b c d\\neq 0 .\n\\]  \nIntroduce  \n\\[\np=-\\dfrac{b}{a},\\qquad q=-\\dfrac{c}{a},\\qquad r=-\\dfrac{d}{a}. \\tag{2}\n\\]\n\nFix an integer \\(n\\ge 2\\).\n\n(i)  Prove that for every \\(n\\) there exist polynomials  \n\\[\nP_{n}(p,q,r),\\;B_{n}(p,q,r),\\;C_{n}(p,q,r)\\in\\mathbb{Q}[p,q,r] \\tag{3}\n\\]  \nsatisfying  \n\\[\nz^{n}=P_{n}+B_{n}\\,z+C_{n}\\,z^{2}. \\tag{4}\n\\]\n\nShow further that these three polynomials are unique provided that the cubic \\((1)\\) is {\\it irreducible} over \\(\\mathbb{Q}\\).\n\n(ii)  Put  \n\\[\nK_{n}:=\\mathbb{Q}(a,b,c,d,z^{n})\n\\]  \nand consider the quadratic over \\(K_{n}\\)\n\\[\nQ_{n}(T)=C_{n}\\,T^{2}+B_{n}\\,T+\\bigl(P_{n}-z^{n}\\bigr). \\tag{5}\n\\]\n\n1.  Prove that \\(Q_{n}(z)=0\\) and deduce \\([K_{n}(z):K_{n}]\\le 2\\).\n    (Explain separately why the statement is immediate when \\(C_{n}=0\\).)\n\n2.  Assuming \\(C_{n}\\neq 0\\), derive the quadratic–formula identity  \n\\[\nz=\\dfrac{-B_{n}\\pm\\sqrt{\\Delta_{n}}}{2C_{n}},\\qquad \n\\Delta_{n}=B_{n}^{2}+4C_{n}(z^{n}-P_{n})\\in K_{n}. \\tag{6}\n\\]\n\n3.  Assume from now on that  \n\n   •  the cubic \\((1)\\) is irreducible over \\(\\mathbb{Q}\\);  \n   •  its discriminant \\(\\Delta\\) is not a square in \\(\\mathbb{Q}\\)\n     (equivalently, the splitting–field Galois group is \\(S_{3}\\)).\n\n   Let \\(z_{1}=z,z_{2},z_{3}\\) be the roots of \\((1)\\) and define  \n   \\[\n   \\mu:=\\dfrac{z_{2}}{z_{1}},\\qquad \n   \\nu:=\\dfrac{z_{3}}{z_{1}},\\qquad \n   R:=\\{1,\\mu,\\nu,\\mu^{-1},\\nu^{-1},\\mu/\\nu\\}. \\tag{7}\n   \\]\n\n   (a)  Prove that  \n        \\[\n        [\\mathbb{Q}(\\mu):\\mathbb{Q}]=[\\mathbb{Q}(\\nu):\\mathbb{Q}]\\in\\{2,3,6\\}.\n        \\]\n        Show further that if one of \\(\\mu,\\nu\\) is a root of unity, then its\n        order belongs to the set \\(\\{1,2,3,4,6\\}\\).\n\n   (b)  For\n        \\[\n          H_{n}:=\\{\\tau\\in\\operatorname{Gal}(L/\\mathbb{Q})\\mid\n          (\\tau(z)/z)^{\\,n}=1\\}\n        \\]\n        prove the chain of equivalences  \n        \\[\n        Q_{n}(T)\\text{ is irreducible over }K_{n}\n        \\;\\Longleftrightarrow\\;\n        z\\notin K_{n}\n        \\;\\Longleftrightarrow\\;\n        R\\setminus\\{1\\}\\text{ contains an }n\\text{-th root of unity}. \\tag{8}\n        \\]\n\n   (c)  Deduce:\n\n        •  If no element of \\(R\\setminus\\{1\\}\\) is a root of unity, then\n          \\(Q_{n}(T)\\) splits over \\(K_{n}\\) and \\(z\\in K_{n}\\) for every\n          \\(n\\ge 2\\).\n\n        •  If the subgroup \\(\\langle\\mu,\\nu\\rangle\\subset L^{\\times}\\) is\n          finite of order \\(m\\in\\{2,3,4,6\\}\\), then \\(Q_{n}(T)\\) is irreducible\n          over \\(K_{n}\\) exactly for those integers \\(n\\) that share a common\n          divisor with \\(m\\) (equivalently, for which \\(\\gcd(m,n)\\neq1\\)),\n          and it splits otherwise.  Consequently \\(z\\) can be expressed as a\n          rational function in \\(z^{n}\\) and \\(a,b,c,d\\) precisely when\n          \\(\\gcd(m,n)=1\\).\n\n(b)  For every non–zero complex number \\(x\\) put  \n\\[\nu=x+\\dfrac{1}{x}.\n\\]\nProve the inversion identity  \n\\[\nx=\\dfrac{x^{7}+u^{5}-4u^{3}+3u}{\\,u^{6}-5u^{4}+6u^{2}-1\\,} \\tag{9}\n\\]\nand verify that the denominator is not the zero polynomial, so the\nformula is valid for all \\(x\\) with\n\\(u^{6}-5u^{4}+6u^{2}-1\\neq 0\\).\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "solution": "(a)  \n\n(i)  Divide \\((1)\\) by \\(a\\) to obtain the monic relation  \n\\[\nz^{3}=p z^{2}+q z+r. \\tag{10}\n\\]\nPut  \n\\[\n\\mathcal{V}:=\\operatorname{Span}_{\\mathbb{Q}}\\{1,z,z^{2}\\}.\n\\]\nBecause \\((10)\\) reduces every power \\(z^{k}\\;(k\\ge 3)\\) to an element of\n\\(\\mathcal{V}\\), there certainly exist polynomials\n\\(P_{n},B_{n},C_{n}\\in\\mathbb{Q}[p,q,r]\\) satisfying \\((4)\\).\n\nAssume now that the cubic \\((1)\\) is irreducible over \\(\\mathbb{Q}\\); then\n\\(\\deg_{\\mathbb{Q}}(z)=3\\) and the family \\(\\{1,z,z^{2}\\}\\) is\n\\(\\mathbb{Q}\\)-linearly independent.  Hence the representation in \\((4)\\)\nis unique: if  \n\\(P_{n}+B_{n}z+C_{n}z^{2}=P_{n}^{\\prime}+B_{n}^{\\prime}z+\nC_{n}^{\\prime}z^{2}\\),  \nthen \\((P_{n}-P_{n}^{\\prime})+(B_{n}-B_{n}^{\\prime})z+\n(C_{n}-C_{n}^{\\prime})z^{2}=0\\) and independence forces\n\\(P_{n}=P_{n}^{\\prime},\\,B_{n}=B_{n}^{\\prime},\\,C_{n}=C_{n}^{\\prime}\\).\n\nTo compute these polynomials one may proceed inductively.  Assume  \n\\[\nz^{k}=P_{k}+B_{k}z+C_{k}z^{2},\\qquad k\\ge 0. \\tag{11}\n\\]\nMultiplying by \\(z\\) and reducing the third power with \\((10)\\) shows  \n\\[\nz^{k+3}=p z^{k+2}+q z^{k+1}+r z^{k}. \\tag{12}\n\\]\nTaking coefficients of \\(1,z,z^{2}\\) we obtain  \n\\[\n\\begin{pmatrix}P_{k+3}\\\\[2pt] B_{k+3}\\\\[2pt] C_{k+3}\\end{pmatrix}\n=\np\\begin{pmatrix}P_{k+2}\\\\ B_{k+2}\\\\ C_{k+2}\\end{pmatrix}\n+\nq\\begin{pmatrix}P_{k+1}\\\\ B_{k+1}\\\\ C_{k+1}\\end{pmatrix}\n+\nr\\begin{pmatrix}P_{k}\\\\ B_{k}\\\\ C_{k}\\end{pmatrix}. \\tag{13}\n\\]\nWith the initial triples \\((1,0,0),(0,1,0),(0,0,1)\\) this triangular\nrecurrence determines uniquely the three sequences\n\\((P_{k})_{k\\ge 0},(B_{k})_{k\\ge 0},(C_{k})_{k\\ge 0}\\) in\n\\(\\mathbb{Q}[p,q,r]\\).  Setting \\(k=n\\) gives \\((4)\\).\n\n%---------------------------------------------------------------------------  \n(ii-1)  Substituting \\(T=z\\) in \\(Q_{n}(T)\\) shows \\(Q_{n}(z)=0\\), so\n\\([K_{n}(z):K_{n}]\\le 2\\).\n\nIf \\(C_{n}=0\\) then \\(Q_{n}(T)\\) is linear and yields\n\\[\nz=\\dfrac{z^{\\,n}-P_{n}}{B_{n}}\\in K_{n},\n\\]\nhence \\([K_{n}(z):K_{n}]=1\\).  From now on we treat the generic case\n\\(C_{n}\\neq 0\\).\n\n(ii-2)  When \\(C_{n}\\neq 0\\) the quadratic formula applied to\n\\(Q_{n}(T)=0\\) gives \\((6)\\).\n\n%---------------------------------------------------------------------------  \n(ii-3)  Throughout the remainder of part (a) we assume that the cubic\n\\((1)\\) is irreducible and that \\(\\Delta\\notin\\mathbb{Q}^{2}\\); hence the\nsplitting field \\(L\\) of \\((1)\\) satisfies\n\\(\\operatorname{Gal}(L/\\mathbb{Q})\\cong S_{3}\\).\n\nWrite \\(z_{1}=z,z_{2},z_{3}\\) for the three roots and define\n\\(\\mu,\\nu,R\\) as in \\((7)\\).\n\n(α)  {\\it Degrees of \\(\\mu,\\nu\\) and possible orders.}  \nBecause \\(\\mu,\\nu\\in L\\), their degrees divide\n\\(|\\operatorname{Gal}(L/\\mathbb{Q})|=6\\).  The stabiliser of \\(\\mu\\) in\n\\(S_{3}\\) is one of\n\\(\\{1\\},\\langle(1\\,2)\\rangle,\\langle(1\\,2\\,3)\\rangle\\),\nhaving indices \\(6,3,2\\) respectively, so\n\\([\\mathbb{Q}(\\mu):\\mathbb{Q}]\\in\\{6,3,2\\}\\); the same holds for\n\\(\\nu\\).  Concrete examples such as\n\\(x^{3}-x+1,\\;x^{3}-3x+1,\\;x^{3}-3x^{2}+1\\) exhibit the three\npossibilities.\n\nAssume \\(\\mu\\) (or \\(\\nu\\)) is a root of unity of order \\(m\\).\nThen \\(\\mathbb{Q}(\\mu)\\subseteq\\mathbb{Q}(\\zeta_{m})\\) and so\n\\(\\varphi(m)=[\\mathbb{Q}(\\zeta_{m}):\\mathbb{Q}]\\) divides \\(6\\).\nBesides the values \\(1,2,3,4,6\\) the divisibility \\(\\varphi(m)\\mid6\\)\nalso allows \\(m=7,9,14,18\\).  The cyclotomic fields of degree \\(6\\) are\nGalois and have cyclic Galois groups, which cannot be embedded into\nthe non-abelian group \\(S_{3}\\); hence these four possibilities are\nimpossible, and the claimed list of orders follows.\n\n(β)  {\\it The subgroup fixing \\(z^{n}\\).}  \nFor \\(\\tau\\in\\operatorname{Gal}(L/\\mathbb{Q})\\) put  \n\\[\n\\rho(\\tau):=\\dfrac{\\tau(z)}{z}\\in R .  \\tag{14}\n\\]\nBecause \\(\\tau(z^{\\,n})=\\rho(\\tau)^{\\,n}z^{\\,n}\\),  \n\\[\nH_{n}=\\{\\tau\\in \\operatorname{Gal}(L/\\mathbb{Q})\\mid\n       \\rho(\\tau)^{\\,n}=1\\}. \\tag{15}\n\\]\n\n(γ)  {\\it Irreducibility of \\(Q_{n}(T)\\).}  \nSince \\(Q_{n}(z)=0\\),\n\\[\nQ_{n}\\text{ splits over }K_{n}\\;\\Longleftrightarrow\\;z\\in K_{n}.\n\\]\nIf \\(z\\notin K_{n}\\) then \\([K_{n}(z):K_{n}]=2\\) and \\(Q_{n}(T)\\) is\nirreducible.  Relation \\((15)\\) yields  \n\\[\nQ_{n}(T)\\text{ irreducible over }K_{n}\n\\;\\Longleftrightarrow\\;\n\\exists\\tau\\in S_{3}\\text{ with }\\rho(\\tau)\\neq1,\\;\n\\rho(\\tau)^{\\,n}=1\n\\;\\Longleftrightarrow\\;\nR\\setminus\\{1\\}\\text{ contains an }n\\text{-th root of unity}, \\tag{16}\n\\]\nwhich is precisely the chain of equivalences \\((8)\\).\n\n(δ)  {\\it Consequences.}\n\n•  If none of the five non-trivial elements of \\(R\\) is a root of unity,\n  then condition \\((16)\\) never occurs; thus \\(Q_{n}(T)\\) always splits\n  and \\(z\\in K_{n}\\) for every \\(n\\ge 2\\).\n\n•  Suppose \\(\\langle\\mu,\\nu\\rangle\\) is finite of order\n  \\(m\\in\\{2,3,4,6\\}\\).  Every element of \\(R\\setminus\\{1\\}\\) has order\n  dividing \\(m\\).  Consequently\n  \\(R\\setminus\\{1\\}\\) contains an \\(n\\)-th root of unity\n  if and only if \\(\\gcd(m,n)\\neq1\\).\n  Via \\((16)\\) we obtain:  \n\n  \\[\n  Q_{n}(T)\\text{ is irreducible over }K_{n}\n  \\Longleftrightarrow \\gcd(m,n)\\neq1 .\n  \\]\n\n  In the splitting case (\\(\\gcd(m,n)=1\\)) the relation\n  \\(z=\\bigl(-B_{n}\\pm\\sqrt{\\Delta_{n}}\\bigr)/(2C_{n})\\) expresses \\(z\\)\n  as a rational function of \\(z^{\\,n}\\) and \\(a,b,c,d\\); when\n  \\(\\gcd(m,n)\\neq1\\) this is impossible because \\(z\\notin K_{n}\\).\n\n%---------------------------------------------------------------------------  \n(b)  \n\nStarting from the relation  \n\\[\nx^{2}=u\\,x-1, \\tag{17}\n\\]\nwe repeatedly multiply by \\(x\\) and reduce every time with \\((17)\\):\n\n\\[\n\\begin{aligned}\nx^{3}&=(u^{2}-1)x-u,\\\\\nx^{4}&=(u^{3}-2u)x-(u^{2}-1),\\\\\nx^{5}&=(u^{4}-3u^{2}+1)x-u(u^{2}-2),\\\\\nx^{6}&=(u^{5}-4u^{3}+3u)x-(u^{4}-3u^{2}+1),\\\\\nx^{7}&=(u^{6}-5u^{4}+6u^{2}-1)x-(u^{5}-4u^{3}+3u). \\tag{18}\n\\end{aligned}\n\\]\nSolving the last line of \\((18)\\) for \\(x\\) gives the inversion identity\n\\[\nx=\\dfrac{x^{7}+u^{5}-4u^{3}+3u}{u^{6}-5u^{4}+6u^{2}-1},\n\\]\nwhich coincides with \\((9)\\).  The denominator is the non-zero\npolynomial \\(u^{6}-5u^{4}+6u^{2}-1\\); hence the formula is valid for all\n\\(x\\) for which this denominator does not vanish.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.489619",
        "was_fixed": false,
        "difficulty_analysis": "•  The original problem dealt with a quadratic and used a single power zⁿ.\n   Here we move to a *cubic*, which raises the ambient field degree from 2\n   to 3 and forces one to manipulate *three* coefficient–dependent sequences\n   simultaneously instead of one.  \n•  Part (a) demands an explicit construction of Pₙ,Bₙ,Cₙ, the derivation of a\n   pair of coupled linear equations, and the use of Cramer’s rule; none of\n   these is needed for the quadratic case.  The proof also has to guarantee\n   that some small n works for *all* cubics, invoking a non–trivial argument\n   on the coefficients.  \n•  Part (b) asks for the recovery of x from (x⁵,u).  The exponent 5 forces\n   three rounds of substitutions (compare the single round required when the\n   given power is x⁴ in the current kernel variant), and the resulting\n   polynomials are of much higher degree (degree 4 in u).  Careful algebra is\n   indispensable to avoid sign and coefficient errors.  \n•  Together, the two parts blend linear–recurrence theory, determinant\n   techniques, and clever algebraic manipulation, going noticeably beyond the\n   elementary pattern-matching that suffices for the quadratic prototype."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}