Question

Let M/F and K/F be Galois extensions with Galois groups G = Gal(M/F) and H = Gal(K/F). Since M/F is Galois, and K/F is a field extension, we have the composite extension field K M.

Show that K M/F is a Galois extension

Answer #1

Let M/F and K/F be Galois extensions with Galois groups G =
Gal(M/F) and H = Gal(K/F). Since M/F is Galois, and K/F is a field
extension, we have the composite extension field K M
Show that the image is {(τ, υ) ∈ G × H | τ |K∩M =
υ|K∩M }

Let E/F be a finite Galois extension such that Gal(E/F) is
abelian. Prove that for
every intermediate field K, the extension K/F is Galois.

Let H, K be two groups and G = H × K. Let H = {(h, e) | h ∈ H},
where e is the identity in K. Show that G/H is isomorphic to K.

Let f : G → H be a group isomorphism, and K ⊂ G be a subgroup.
Show that f(K) ⊂ H is a subgroup.

f H and K are subgroups of a group G, let (H,K) be the subgroup
of G generated by the elements {hkh−1k−1∣h∈H, k∈K}.
Show that :
H◃G if and only if (H,G)<H

Let N and H be groups, and here for a homomorphism f:
H --> Aut(N) = group automorphism,
let N x_f H be the corresponding semi-direct product.
Let g be in Aut(N), and k be in Aut(H), Let C_g:
Aut(N) --> Aut(N) be given by
conjugation by g.
Now let z := C_g * f * k: H --> Aut(N), where *
means composition.
Show that there is an isomorphism
from Nx_f H to Nx_z H, which takes the natural...

Let E/F be an algebraic extension. Let K and L be intermediate
fields (i.e. F ⊆ K ⊆ E and F ⊆ L ⊆ E).
(i) Prove that if the extension K/F is separable then the
extension KL/L is separable.
(ii) Prove that if the extension K/F is normal then the
extension KL/L is normal.
Note: To make things easier for you, you can assume that E/F is
finite (hence all extensions are finite),

Let G and H be groups, and let
G0 = {(g, 1) : g ∈ G} .
(a) Show that G0 ≅ G.
(b) Show that G0 is a normal subgroup of G × H.
(c) Show that (G × H)/G0 ≅ H.

Let G and H be groups and f:G--->H be a surjective
homomorphism. Let J be a subgroup of H and define f^-1(J) ={x is an
element of G| f(x) is an element of J}
a. Show ker(f)⊂f^-1(J) and ker(f) is a normal subgroup of
f^-1(J)
b. Let p: f^-1(J) --> J be defined by p(x) = f(x). Show p is
a surjective homomorphism
c. Show the set kef(f) and ker(p) are equal
d. Show J is isomorphic to f^-1(J)/ker(f)

Let F⊆K⊆E be extension fields. If K is an algebraic extension of
F and let α∈E be algebraic over K. Show that α is also algebraic
over F.

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