If K is finite and F is an algebraic closuer of K, then the Galois group...

If K is finite and F is an algebraic closuer of K, then the Galois group Aut F over K is abelian. Every element of Aut F over K has infinite order.

Expert Solution

Colby Messinger answered 2 years ago

Let M/F and K/F be Galois extensions with Galois groups G = Gal(M/F) and H =...

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 σ → (σ|M , σ|K) is a homomorphism from Gal(K M/F) to G × H, and that it is one-to-one. [As in the notes, σ|X means the restriction of the map σ to the subset X of its domain.]

Theorem: Let K/F be a field extension and let a ∈ K be algebraic over F....

Theorem: Let K/F be a field extension and let a ∈ K be algebraic over F. If deg(mF,a(x)) = n, then 1. F[a] = F(a). 2. [F(a) : F] = n, and 3. {1, a, a2 , ..., an−1} is a basis for F(a).

Let E/F be an algebraic extension and let K and L be intermediate fields (i.e. F...

Let E/F be an algebraic extension and let K and L be intermediate fields (i.e. F ⊆ K ⊆ E and F ⊆ L ⊆ E). Assume that [K : F] and [L : F] are finite and that at least K/F or L/F is Galois. Prove that [KL : F] = [K : F][L : F] / [K ∩ L : F] .

7.What was Galois’ contribution to the theory of algebraic equations and what were some of the...

7.What was Galois’ contribution to the theory of algebraic equations and what were some of the reasons why it was so remarkable?

Let F be a finite field. Prove that the multiplicative group F*,x) is cyclic.

Let k ⊂ K be an extension of fields (not necessarily finite-dimensional). Suppose f, g ∈...

Let k ⊂ K be an extension of fields (not necessarily finite-dimensional). Suppose f, g ∈ k[x], and f divides g in K[x] (that is, there exists h ∈ K[x] such that g = fh. Show that f divides g in k[x].

The question is: Let G be a finite group, H, K be normal subgroups of G,...

The question is: Let G be a finite group, H, K be normal subgroups of G, and H∩K is also a normal subgroup of G. Using Homomorphism theorem ( or First Isomorphism theorem) prove that G/(H∩K) is isomorphism to a subgroup of (G/H)×(G/K). And give a example of group G with normal subgroups H and K such that G/(H∩K) ≆ (G/H)×(G/K), with explanation. I was trying to find some solutions for the isomorphism proof part, but they all seems to...

Show that if F is a finite extension of Q, then the torsion subgroup of F*...

Show that if F is a finite extension of Q, then the torsion subgroup of F* is finite

Use the function below to answer parts a-k. f(x)=x^3-5x^2+3x+9 a. Is the function algebraic or exponential?...

Use the function below to answer parts a-k. f(x)=x^3-5x^2+3x+9 a. Is the function algebraic or exponential? b. Find the domain of the function. c. Find the interval where the function is continuous. If the function is discontinuous at a value, specify the first continuity condition that is not satisfied. d. What is/are the x-intercepts? e. What is the y-intercept? f. Find all of its critical values stating if they are absolute/relative max or min. g. What are the hypercritical values?...

1.The algebraic minterm of F(A,B,C) = ABC + B’C is: 2.The digital minterm of F is:...

1.The algebraic minterm of F(A,B,C) = ABC + B’C is: 2.The digital minterm of F is: 3. The digital maxterm for F is: 4.The digital minterm of F’ is: 5.The digital maxterm of F’ is: 6.If the digital minterm of F(A,B,C,D) = Σ m(0, 2, 4, 6), the digital maxterm is: 7.The minimum sum of products for F(A,B,C) = Σ m(0, 1, 4) is:

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