Question

In: Advanced Math

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.

Expert Solution

Given data,

Let F be a finite field

Let char(F) = m

Then

,

In particular,

Hence,

=

upto m times

Therefore, the asked n is of the form n = mK, .

Let char(F) =

If possible suppose that be not prime

Let = rs,(a conjugate number where r,s > 1).

So, .a = (by definition of characteristic)

In particular, =

=> =

=>

=> or

(as F is a field, hence it is a integral domain)

char(F) = r < or char(F) = S <

- a contradiction.

(as is the least positive integer such that = )

Hence, char(F) = p, a prime.

Colby Messinger answered 1 year ago

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

Let {an}n∈N be a sequence with lim n→+∞ an = 0. Prove that there exists a...

Let {an}n∈N be a sequence with lim n→+∞ an = 0. Prove that there exists a subsequence {ank }k∈N so that X∞ k=1 |ank | ≤ 8

Let t be a positive integer. Prove that, if there exists a Steiner triple system of...

Let t be a positive integer. Prove that, if there exists a Steiner triple system of index 1 having v varieties, then there exists a Steiner triple system having v^t varieties

Let n be a positive integer. Prove that if n is composite, then n has a...

Let n be a positive integer. Prove that if n is composite, then n has a prime factor less than or equal to sqrt(n) . (Hint: first show that n has a factor less than or equal to sqrt(n) )

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.

Question 1. Let V and W be finite dimensional vector spaces over a field F with...

Question 1. Let V and W be finite dimensional vector spaces over a field F with dimF(V ) = dimF(W) and let T : V → W be a linear map. Prove there exists an ordered basis A for V and an ordered basis B for W such that [T] A B is a diagonal matrix where every entry along the diagonal is either a 0 or a 1. Hint 1. Suppose A = {~v1, . . . , ~vn}...

Let F be a field and R = Mn(F) the ring of n×n matrices with entires...

Let F be a field and R = Mn(F) the ring of n×n matrices with entires in F. Prove that R has no two sided ideals except (0) and (1).

Prove a (Dedekind) set is inﬁnite iff there exists an injective function f : N→ A....

Prove a (Dedekind) set is inﬁnite iff there exists an injective function f : N→ A. please help prove this clearly and i will rate the best answer thank you