Galois Theory Problem Sets

Problem Set 1

Here's a start:  1.6, 3..8, 3.21: Show by example that if hypothesis (1) is deleted from the statement of Eisenstein's Criterion, the resulting statement is false.  

Here's an end:  4.4, 4.6, 4.7, 4.9.

I think that's pretty likely to stay as PS1. 

Problem Set 2

Probably not a bad time for me to start working on this.  What about chapter 5?  How about 5.3 and 5.7?  The second might not make the cut, but I think the first is here to stay.  Chapter 6?  6.14, 6.16, 6.20 (using 6.16):  Show that every subextension (use the natural definition) of a simple algebraic extensions over an infinite field is simple.

I think I'm keeping everything above.  To that add 8.1, 8.11, and 8.14.  8.14 is kinda long to typeset.  I'll try.  Please ask me if there are questions about what it means.  8.14  Let L:K be any field extension.  (a)  Suppose H1 and H2 are subgroups of the Galois group of L over K, and let    H1 v H2 denote the subgroup of the Galois group generated by the union of these two groups.  Show that (H1 v H2)-dagger = the intersection between H1-dagger and H2-dagger.  (b)  Suppose M1 and M2 are intermediate fields of L:K, and let M1M2 denote the intermediate field generated by the union of these two fields.  Show that (M1M2)* = the intersection of M1* and M2*.

I think that's eight problems.  Please make sure you do too.

Problem Set 3

9.6, 9.8, 10.3,  This addresses one of Dan's questions:  10.6  Show that if G is an infinite group of automorphisms of a field K, then [K:G-dagger] is infinite. 

One from chapter 11.  In Theorem 11.12, Stewart says that [L:K]=[L:K0] and K a subset of K0 implies K = K0.  Justify this.  If you used anything particular to fields, does the generalisation to vector spaces also hold?  (I mean, if the dimension of V over F1 is equal to the dimension of V over F2, and F1 is a subset of F2, is it true that F1=F2?)  Either prove or refute.  

11.8  Suppose L:K is a normal algebraic extension, E any field containing L, and A any subset of E.  Show that the Galois group of L(A):K(A) is isomorphic to a subgroup of the Galois group of L:K, the isomorphism being given by restricting elements of the Galois group of L(A):K(A) to L.  (You should begin by indicating why this restriction operation takes elements of the Galois group of L(A):K(A) to elements of the Galois group of L:K.)  At least to me - the interesting part of the question - show by example that this result becomes false if L:K is not assumed normal.  

Aside from that, also please complete 11.5.  I think it's likely that 12.2 will be added.   How about this promise?  Eventually I will collect 12.2, so completing it is a good thing to do.  I may end up deciding that we're not ready for it to be including in Dan's birthday present.   

I think this is 8 (maybe 7) questions.  I hope you do too.  


Problem Set 4

13.8 (including 12.5),

14.0 This should be easy:  prove for any element x in a group G, the centraliser of x in G is a subgroup.

14.1 and 8 as one question.

15.3, 15.6, 15.10, 15.11 (it seems I might give up on some of these in chapter 15)

complete either 7.5 or 7.13 (notice that hints continue on the next page for 7.13)

Ok, I think I'm stopping and keeping all from 15.  It was an important chapter.  But, I won't be adding anything for 16 or 24.  And I think therefore you may choose to prepare those chapters for the final, or you may not.  As you wish.  We'll probably discuss this.