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

Expert Solution

Please feel free to ask any query in the comment box and don't forget to rate if you like

Colby Messinger answered 2 years ago

Let F be a finite field. Prove that there exists an integer n≥1, such that n.1_F...

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

Let G be a cyclic group generated by an element a. a) Prove that if an...

Let G be a cyclic group generated by an element a. a) Prove that if an = e for some n ∈ Z, then G is finite. b) Prove that if G is an infinite cyclic group then it contains no nontrivial finite subgroups. (Hint: use part (a))

Let F be a field. (a) Prove that the polynomials a(x, y) = x^2 − y^2,...

Let F be a field. (a) Prove that the polynomials a(x, y) = x^2 − y^2, b(x, y) = 2xy and c(x, y) = x^2 + y^2 in F[x, y] form a Pythagorean triple. That is, a^2 + b^2 = c^2. Use this fact to explain how to generate right triangles with integer side lengths. (b) Prove that the polynomials a(x,y) = x^2 − y^2, b(x,y) = 2xy − y^2 and c(x,y) = x^2 − xy + y2 in F[x,y]...

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.

Prove the following: (a) Let A be a ring and B be a field. Let f...

Prove the following: (a) Let A be a ring and B be a field. Let f : A → B be a surjective homomorphism from A to B. Then ker(f) is a maximal ideal. (b) If A/J is a field, then J is a maximal ideal.

Prove the theorem in the lecture:Euclidean Domains and UFD's Let F be a field, and let...

Prove the theorem in the lecture:Euclidean Domains and UFD's Let F be a field, and let p(x) in F[x]. Prove that (p(x)) is a maximal ideal in F[x] if and only if p(x) is irreducible over F.

Let f(x)∈F[x] be separable of degree n and let K be the splitting field of f(x)....

Let f(x)∈F[x] be separable of degree n and let K be the splitting field of f(x). Show that the order of Gal(K/F) divides n!.

2 Let F be a field and let R = F[x, y] be the ring of...

2 Let F be a field and let R = F[x, y] be the ring of polynomials in two variables with coefficients in F. (a) Prove that ev(0,0) : F[x, y] → F p(x, y) → p(0, 0) is a surjective ring homomorphism. (b) Prove that ker ev(0,0) is equal to the ideal (x, y) = {xr(x, y) + ys(x, y) | r,s ∈ F[x, y]} (c) Use the first isomorphism theorem to prove that (x, y) ⊆ F[x, y]...

abstract algebra Let G be a finite abelian group of order n Prove that if d...

abstract algebra Let G be a finite abelian group of order n Prove that if d is a positive divisor of n, then G has a subgroup of order d.

9. Let f be continuous on [a, b]. Prove that F(x) := sup f([x, b]) is...

9. Let f be continuous on [a, b]. Prove that F(x) := sup f([x, b]) is continuous on [a, b]

Question