# Cogalois Theory by Toma Albu

By Toma Albu

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6. (3) * (4), (4) a ( I ) , and (5) u (6) are obvious. (3) a (5): Let T = { ti I i E I ) be a set of representatives of G I F * . Then T is a basis for E over F . Let M = { x j I j E J) C G be such - that ;x # ;x for each j l # j2 in J . If { x l , . . ,x,,) is a finite subset of M, then we have G = t i , , . . ,x,, = t i n , with i l , . . ,i, E I. It foll o w ~that X I = Xlti,,. . ,X, = Xr1ti, for some X I ? . ,A,, E F * . The set {ti,, . . , t i n ) is clearly linearly independent over F , since it is a subset of the basis T.

3. Find [Q( ' 14. 12 t o prove Exercise 12 (b). 4. Let F be a field satisfying the condition Co(n;a ) , and let m = [ F ( f i ) : F]. Show that the map a : ID,, -+ Intermediate ( F ( *)IF)), a ( d ) = F ( %d ) , establishes an anti-isomorphism of lattices. 15. Find 5 . Find all subfields of the field Q( fi) , where n E N* and a E q. 6. Let p E IP and let F be a field such that p,(F) = (1). Prove that F satisfies the condition Cl (p ; a ) for any a E F*. 51 for n 6 4 . 16. Show that if G is a finite cyclic group of order n , then OG = ID,,.

0 -4pplications of the Kneser Criterion to algebraic number fields will be given in Chapter 9. 3. Exercises to Chapter 2 1. Prove that pn ( F( X I , . . ,X,)) m,nEN*. = pn ( F ) for any field F and any 2. Let p be any positive prime number, and let n be any positive integer. Prove that the extension IF', ( X I , . . ,Xn)/IFp ( X r , . . ,X,P) is an extension of degree p7" with Cogalois group isomorphic t o a countably infinite direct sum of copies of the cyclic group Z,. 3. Show that Cog(F4/IF2) 2 K.