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.

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Colby Messinger answered 10 months ago

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 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.

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]

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

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