proof that the compositum of a Galois extension and another extension is Galois
To see that is Galois, note that since is Galois, is a splitting field of a set of polynomials over ; clearly is a splitting field of the same set of polynomials over . Also, if is separable over , then also is separable over . Thus is normal and separable over , so is Galois. is obviously Galois over since .
Let be the restriction map
is clearly a group homomorphism, and since is normal over , is well-defined.
Now, the image of is a subgroup of with fixed field , and thus the image of is . Claim . is obvious: any element is fixed by for each since fixes . Thus . To see the reverse inclusion, choose ; then is fixed by each for . But , so that (as an element of ), is fixed by each . Thus so that .
Thus , and is then an isomorphism . ∎
- 1 Morandi, P., Field and Galois Theory, Springer, 1996.
|Title||proof that the compositum of a Galois extension and another extension is Galois|
|Date of creation||2013-03-22 18:41:58|
|Last modified on||2013-03-22 18:41:58|
|Last modified by||rm50 (10146)|