Field Theory and Its Classical Problems, Volume 14
Field Theory and its Classical Problems lets Galois theory unfold in a natural way, beginning with the geometric construction problems of antiquity, continuing through the construction of regular n-gons and the properties of roots of unity, and then on to the solvability of polynomial equations by radicals and beyond. The logical pathway is historic, but the terminology is consistent with modern treatments. No previous knowledge of algebra is assumed. Notable topics treated along this route include the transcendence of e and π, cyclotomic polynomials, polynomials over the integers, Hilbert's irreducibility theorem, and many other gems in classical mathematics. Historical and bibliographical notes complement the text, and complete solutions are provided to all problems.
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algebraic extension algebraic over F algebraically independent angle automorphism calculation complex numbers conjugates constructible cube cyclic cyclotomic polynomial defined denote divide division algorithm element of F example exists exponents extension of F F0 c F factors field F finite extension finite number fixed field follows G F[x Galois group given hence integer integral coefficients intermediate fields irreducible over Q irreducible polynomial Let f linear mapping mathematical minimal polynomial multiplication nomial nontrivial normal extension normal subgroup obtain one-to-one permutation polynomial equation polynomial of degree polynomial over Q polynomial with integral power series previous problem primitive nth roots primitive root modulo proof of Theorem quadratic extension rational number rational roots regular n-gon relatively prime result roots of unity Section sequence of fields smallest field containing solution solvable by radicals splitting field suppose symmetric functions term theory transpositions trisect values variables write