Question

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.

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 n be a positive integer and let U be a finite subset of
Mn×n(C) which is closed under multiplication of matrices. Show that
there exists a matrix A in U satisfying tr(A) ∈ {1,...,n}

Let n be an integer, with n ≥ 2. Prove by contradiction that if
n is not a prime number, then n is divisible by an integer x with 1
< x ≤√n.
[Note: An integer m is divisible by another integer n if there
exists a third integer k such that m = nk. This is just a formal
way of saying that m is divisible by n if m n is an integer.]

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.

Let f(n) be a negligible function and k a positive integer.
Prove the following:
(a) f(√n) is negligible.
(b) f(n/k) is negligible.
(c) f(n^(1/k)) is negligible.

Prove that for every positive integer n, there exists a multiple
of n that has for its digits only 0s and 1s.

Prove that for every positive integer n, there exists an
irreducible polynomial of degree n in Q[x].

Prove that if for epsilon >0 there exists a positive integer n
such that for all n>N we have p_n is an element of
(x+(-epsilon),x+epsilon) then p_1,p_2, ... p_n converges to
x.

Discrete Math
6. Prove that for all positive integer n, there exists an even
positive integer k such that
n < k + 3 ≤ n + 2
. (You can use that facts without proof that even plus even is
even or/and even plus odd is odd.)

Let
n be a positive integer and let S be a subset of n+1 elements of
the set {1,2,3,...,2n}.Show that
(a) There exist two elements of S that are relatively prime,
and
(b) There exist two elements of S, one of which divides the
other.

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