Question

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.

Answer #1

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 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

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.

Let
E/F be a field extension, and let α be an element of E that is
algebraic over F.
Let p(x) = irr(α, F) and n = deg p(x).
(a) For f(x) ∈ F[x], let r(x) (∈ F[x]) be the remainder of
f(x) when divided by p(x).
Prove that f(x) +p(x)= r(x)+p(x)in F[x]/p(x).
(b) Prove that if |F| < ∞, then | F[x]/p(x)| = |F|n. (For a
set A, we denote by |A| the number of elements in A.)

For an abelian group G, let tG = {x E G: x has finite order}
denote its torsion subgroup.
Show that t defines a functor Ab -> Ab if one defines t(f) =
f|tG (f restricted on tG) for every homomorphism f.
If f is injective, then t(f) is injective.
Give an example of a surjective homomorphism f for which t(f)
is not surjective.

Let G be an Abelian group. Let k ∈ Z be nonzero. Define φ : G →
G by φ(x) = x^ k . (a) Prove that φ is a group homomorphism. (b)
Assume that G is finite and |G| is relatively prime to k. Prove
that Ker φ = {e}.

Let A be a finite set and f a function from A to A.
Prove That f is one-to-one if and only if f is onto.

4) Let F be a finite field. Prove that there exists an integer n
≥ 1, such that n.1F = 0F . Show further that the smallest positive
integer with this property is a prime number.

Prove that if F is a field and K = FG for a finite
group G of automorphisms of F, then there are only finitely many
subfields between F and K.
Please help!

Abstract Algebra (Modern Algebra)
Prove that every subgroup of an abelian group is abelian.

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