7. Let E be a ﬁnite extension of the ﬁeld F of prime characteristic p. Show...

7. Let E be a ﬁnite extension of the ﬁeld F of prime characteristic p. Show that the
extension is separable if and only if E = F(Ep).

Expert Solution

Colby Messinger answered 2 years ago

let E be a finite extension of a ﬁeld F of prime characteristic p, and let...

let E be a finite extension of a ﬁeld F of prime characteristic p, and let K = F(Ep) be the subﬁeld of E obtained from F by adjoining the pth powers of all elements of E. Show that F(Ep) consists of all ﬁnite linear combinations of elements in Ep with coeﬃcients in F.

Show that, for any events E and F, P(E ∪ F) = P(E) + P(F) −...

Show that, for any events E and F, P(E ∪ F) = P(E) + P(F) − P(E ∩ F). Only use the probability axioms and indicate which axiom you use where

Let E/F be a field extension. Let a,b be elements elements of E and algebraic over...

Let E/F be a field extension. Let a,b be elements elements of E and algebraic over F. Let m=[Q(a):Q] and n=[Q(b):Q]. Assume that gcd(m,n)=1. Determine the basis of Q(a,b) over Q.

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] .

Problem 3. Let F ⊆ E be a field extension. (i) Suppose α ∈ E is...

Problem 3. Let F ⊆ E be a field extension. (i) Suppose α ∈ E is algebraic of odd degree over F. Prove that F(α) = F(α^2 ). Hints: look at the tower of extensions F ⊆ F(α^2 ) ⊆ F(α) and their degrees. (ii) Let S be a (possibly infinite) subset of E. Assume that every element of S is algebraic over F. Prove that F(S) = F[S]

Let gcd(a, p) = 1 with p a prime. Show that if a has at least...

Let gcd(a, p) = 1 with p a prime. Show that if a has at least one square root, then a has exactly 2 roots. [hint: look at generators or use x^2 = y^2 (mod p) and use the fact that ab = 0 (mod p) the one of a or b must be 0(why?) ]

Let E be an extension field of a finite field F, where F has q elements....

Let E be an extension field of a finite field F, where F has q elements. Let a in E be an element which is algebraic over F with degree n. Show that F(a) has q^n elements. Please provide an unique answer and motivate all steps carefully. I also prefer that the solution is provided as written notes.

Let p be a prime and d a divisor of p-1. show that the d th...

Let p be a prime and d a divisor of p-1. show that the d th powers form a subgroup of U(Z/pZ) of order (p-1)/d. Calculate this subgroup for p=11, d=5; p=17,d=4 ;p=19,d=6

Let E and F be independent events. Show that the events E and F care also...

Let E and F be independent events. Show that the events E and F care also independent. Hint: Start with P(E∩Fc) =P(E)−P(E∩F) and use the independence of E and F.

Let T∈ L(V), and let p ∈ P(F) be a polynomial. Show that if p(λ) is...

Let T∈ L(V), and let p ∈ P(F) be a polynomial. Show that if p(λ) is an eigenvalue of p(T), then λ is an eigenvalue of T. Under the additional assumption that V is a complex vector space, and conclude that {μ | λ an eigenvalue of p(T)} = {p(λ) | λan eigenvalue of T}.

Question