Galois groups of tensor products and fixed fields

[dabien]

Homework Statement

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

#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, $$H \cap N=$${1} and $$HN=G$$. Let $$B_N$$ be the fixed field of N (so $$B_N$$ is Galois over F) and $$B_H$$ be the fixed field of H. Prove that E is isomorphic to $$B_N \otimes B_H$$. (If H is also normal in G then $$G\cong H \times N$$ giving a converse to the preceding)

Homework Equations

The Attempt at a Solution

I am trying to get started...

 

[dabien]

My progress:
For #1:
I note that one equivalent condition for an extension to be Galois is that the number of automorphisms which fix the base field equals the degree of the extension. It is clear that $$G_1 \times G_2$$ is contained in the group of automorphisms of $$B_1 \otimes B_2$$ which fix F. Now I need help trying to argue that any such automorphism is necessarily in $$G_1 \times G_2$$. I also need help trying to check orders to confirm that the extension is Galois.

For 2) I have read a condition for when the tensor product of two fields is actually equal to the product of the fields (when the fields are linearly disjoint). I need help verifying that this holds here. I also need to show that this product has to be all of E.
I have a stated lemma: If H,K are two normal subgroups of a group G such that they intersect trivially and $$HK=G$$, then G is isomorphic to H times K. I need help to show that if $$E=B_N \otimes B_H$$ with H and K satisfying the above, then $$G=H \times K$$.