Foundations of Galois TheoryThe first part explores Galois theory, focusing on related concepts from field theory. The second part discusses the solution of equations by radicals, returning to the general theory of groups for relevant facts, examining equations solvable by radicals and their construction, and concludes with the unsolvability by radicals of the general equation of degree n is greater than 5. 1962 edition. |
Contents
THE ELEMENTS OF FIELD THEORY | 1 |
NECESSARY FACTS FROM THE THEORY OF GROUPS | 16 |
GALOIS THEORY | 29 |
The Galois group of a normal subfield | 39 |
ADDITIONAL FACTS FROM THE GENERAL THEORY OF GROUPS | 45 |
Normal series | 53 |
THE UNSOLVABILITY BY RADICALS OF THE GENERAL EQUATION | 92 |
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Common terms and phrases
a₁ Abelian algebraically generated extension arbitrary field automorphism b₁ belongs called class of extensions clearly coefficients coincides complex numbers composite algebraic extension consider coset Hg cyclic group decomposition field denoted distinct from zero element g epimorphism equation of degree extensions of type fact factor group field K field K(G field of rational field P(x field Q finite extension follows formula fractional power series fundamental field G₁ G₂ Galois group G(K Galois theory group G H₁ H₂ Hence homomorphic image i₁ identity induced intermediate field inverse irreducible polynomial isomorphic j₁ K₂ kernel L₁ leaves invariant minimal polynomial monomorphism morphism n-th root nomial normal divisor normal extension normal field normal radical extension normal series obtain one-one permutation poly polynomial f(x polynomial ƒ rational numbers root of unity simple algebraic extension solvable by radicals solvable group solvable series subfield subgroup H symmetric polynomial t₁ t₂ transformation whole number α₁