- #1

dabien

- 4

- 0

## Homework Statement

Prove

Suppose that [tex]B_1, B_2[/tex] are Galois extensions of F with respective Galois groups [tex]G_1, G_2[/tex], and that [tex]E=B_1 \otimes B_2[/tex] is a field. Then it is Galois over F with Galois group isomorphic to [tex]G_1 \times G_2[/tex].

#2

Suppose that E is a Galois extension of F with Galois group G and that G contains subgroups H and N with N normal in G, [tex]H \cap N=[/tex]{1} and [tex]HN=G[/tex]. Let [tex]B_N[/tex] be the fixed field of N (so [tex]B_N[/tex] is Galois over F) and [tex]B_H[/tex] be the fixed field of H. Prove that E is isomorphic to [tex]B_N \otimes B_H[/tex]. (If H is also normal in G then [tex]G\cong H \times N[/tex] giving a converse to the preceding)

## Homework Equations

## The Attempt at a Solution

I am trying to get started...