Abstract Algebra with Applications , 1st Edition by Karlheinz Spindler

By Karlheinz Spindler

A accomplished presentation of summary algebra and an in-depth remedy of the functions of algebraic concepts and the connection of algebra to different disciplines, corresponding to quantity idea, combinatorics, geometry, topology, differential equations, and Markov chains.

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Example. For n 1, show that each of the roots of f (X) = X n − 1 in C is simple. Solution. We have f ′ (X) = ∂(X n − 1) = nX n−1 , so for any root ζ of f (X), f ′ (ζ) = nζ n−1 ̸= 0. 59. Example. Show that 2i is a multiple root of f (X) = X 4 + 8X 2 + 16. Solution. We have f ′ (X) = 4X 3 +16X. Using Long Division and the Euclidean Algorithm we find that gcd(f (X), f ′ (X)) = X 2 + 4, where 2i is also a root of X 2 + 4. Hence 2i is a multiple root of f (X). In fact, X 4 + 8X 2 + 16 = (X 2 + 4)2 , so this is obvious.

Hence the ℓi mj form a basis of M over K and so [M : K] = rs = [M : L] [L : K]. 28 We will often indicate subextensions in diagrammatic form where larger fields always go above smaller ones and the information on the lines indicates dimensions M✳ [M :L] ✮ ✩ L ✤ [L:K] ✏ ✕ ✚ [M :K]=[M :L] [L:K] K We often suppress ‘composite’ lines such as the dashed one. Such towers of extensions are our main objects of study. We can build up sequences of extensions and form towers of arbitrary length. Thus, if L1 /K, L2 /L1 , .

We build up the list of monomorphisms in stages. √ √ First consider monomorphisms that fix 3 2 and hence fix the subfield Q( 3 2). These form the subset √ √ 3 3 (Q( 2, ζ ), C) ⊆ Mono (Q( 2, ζ3 ), C). MonoQ( √ 3 3 Q 2) √ √ 3 3 We know that Q( 2, ζ3 ) = Q( 2)(ζ3 ) and that ζ3 is a root of the irreducible cyclotomic √ polynomial Φ3 (X) = X 2 + X + 1 ∈ Q( 3 2)[X]. So there are two monomorphisms id, α0 fixing √ Q( 3 2), where α0 has the effect (√ √ ) 3 2 −→ 3 2 α0 : . ζ3 −→ ζ32 √ √ Next we consider monomorphisms that send 3 2 to 3 2 ζ3 .

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