Question

Prove that if E is a finite field with characteristic p, then the number of elements in E equals p^n, for some positive integer n.

Answer #1

Prove that for any positive integer n, a field F can have at
most a finite number of elements of multiplicative order at most
n.

Let E be a field of characteristic p, where p is a prime number.
Show that for all x, y that are elements of E, we have (x + y)^p
=x^p + y^p, and hence by induction, (x + y)^p^n = x^p^n + y^p^n
.

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

4. Prove that if p is a prime number greater than 3, then p is
of the form 3k + 1 or 3k + 2.
5. Prove that if p is a prime number, then n √p is irrational
for every integer n ≥ 2.
6. Prove or disprove that 3 is the only prime number of the form
n2 −1.
7. Prove that if a is a positive integer of the form 3n+2, then
at least one prime divisor...

Let P be a finite poset, and let J(P) be the poset whose
elements are the ideals of P, ordered by inclusion. Prove that P is
a lattice.

Let E/F be a finite Galois extension such that Gal(E/F) is
abelian. Prove that for
every intermediate field K, the extension K/F is Galois.

Let
E/F be a field extension, and let α be an element of E that is
algebraic over F.
Let p(x) = irr(α, F) and n = deg p(x).
(a) For f(x) ∈ F[x], let r(x) (∈ F[x]) be the remainder of
f(x) when divided by p(x).
Prove that f(x) +p(x)= r(x)+p(x)in F[x]/p(x).
(b) Prove that if |F| < ∞, then | F[x]/p(x)| = |F|n. (For a
set A, we denote by |A| the number of elements in A.)

Prove If F is
of characteristic p
and p
divides n, then
there are fewer than n
distinct
nth
roots of unity over
F: in this case the derivative is
identically 0
since
n=0
in F. In fact
every root of
x^n-1
is multiple in this case.
Please write legibly, no blurry pictures and no
cursive.

find jacobson radical of polynomials and prove every finite divison
ring is field

Let p be an odd prime and let a be an odd integer with p not
divisible by a. Suppose that p = 4a + n2 for some
integer n. Prove that the Legendre symbol (a/p) equals 1.

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