By Toma Albu
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Extra resources for Cogalois Theory
Example text
3 or Exercise 18, prove that any finite subgroup of the multiplicative group F* of any field F is cyclic. In particular, is cyclic. for any finite field IFq, the group (Darbi i53], [60]). Let F be a field of characteristic 0, and let f = X n - a E F [ X ] be an irreducible binomial with root u E 0. Let k = m a x i m E EDn ( u n l mE F(Cn) }. Prove that the degree of the splitting field of f over F is [ F(C,) : F] . n l k . CHAPTER 2 KNESER EXTENSIONS Prove that the following statements hold for a finite extension E I F .
If gcd(n, e ( F ) ) = 1, then there exist precisely cp(n) primitive n-th roots of unity over F , where cp(n) is the Euler function of n. If there is no danger of ambiguity about the ground field F , we will simply say primitive n-th root of unity. Note that the standard definition of the notion of primitive root of unity is somewhat different: usually, by a primitive n-th root of unity over F one understands any element in F having order n in the group -* F . Clearly, when gcd(n,e(F)) = 1, then the two definitions coincide, but if Char(F) = p > 0 and pl n , then no primitive n-root of unity in the standard definition exists, while, in our definition, always exists a primitive 11-th root of unity.
We also relate the Krieser Criterion t o the Gay-Vdlea Criterion. ecal1 that throughout this monograph we shall use the following notation: P = { p W~ l p prime), P\ P = {2)) u {4h Recall also that R is a fixed algebraically closed field containing the fixed base field F as a subfield; any considered overfield of F is supposed t o be a subfield of R. , a generator of the cyclic group p,(R). 1 T h e Greither-Harrison Criterion Chapter 3 A fundamental concept in the theory of radical extensions is that of purity, which is somewhat related to that used in Group Theory.