Math 361, Spring 2015, Assignment 13

Carefully define the following terms, then give one example and one non-example of each:

1. Solvable by radicals (you may have trouble coming up with non-examples; see pp. 473-474 if you really want one, though I won't quiz you on this).
2. Automorphism (of an extension $F\rightarrow E$).
3. Galois group (of an extension $F\rightarrow E$).
4. Fixed field (of a subgroup of a Galois group).
5. Meet (of two subgroups).
6. Join (of two subgroups).
7. Product (of two subgroups).

Carefully state the following theorems (you do not need to prove them):

1. Fundamental theorem of Galois theory (this is not in the book in any recognizable form, though it is technically encompassed by Theorem 53.6; I recommend consulting lecture notes for a more concise form of the theorem).
2. Theorem relating products to joins.
3. First isomorphism theorem.
4. Second isomorphism theorem.
5. Third isomorphism theorem.

Solve the following problems:

1. Section 34, problems 1, 3, 5, and 7.

--------------------End of assignment--------------------

Questions:

Anybody have an answer for the Fundamental theorem of Galois theory?--Ian.Morse (talk) 17:42, 18 May 2015 (EDT)

Suppose $F$ is either finite or has characteristic zero, that $f\in F[x]$ is non-constant, and the $F\rightarrow E$ is the splitting field of $f$. Then the map that takes a subgroup of $\mathrm{Gal}(E/F)$ to its fixed field is an inclusion-reversing bijection from the set of subgroups of $\mathrm{Gal}(E/F)$ to the set of subextensions of $F\rightarrow E$. -Steven.Jackson (talk) 18:09, 18 May 2015 (EDT)