Question

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.

Answer #1

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.

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.

Prove that if x+ \frac{1}{x} is integer then x^n+ \frac{1}{x^n}
is also integer for any positive integer n.
KEY NOTE: PROVE BY INDUCTION

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.

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.)

Prove that there is no positive integer n so that 25 < n^2
< 36. Prove this by directly proving the negation.Your proof
must only use integers, inequalities and elementary logic. You may
use that inequalities are preserved by adding a number on both
sides,or by multiplying both sides by a positive number. You cannot
use the square root function. Do not write a proof by
contradiction.

Let n ≥ 1 be an integer. Use the Pigeonhole Principle to prove
that in any set of n + 1 integers from {1, 2, . . . , 2n}, there
are two elements that are consecutive (i.e., differ by one).

Prove that if F is a field and K = FG for a finite
group G of automorphisms of F, then there are only finitely many
subfields between F and K.
Please help!

Let F be a field. It is a general fact that a finite subgroup G
of (F^*,X) of the multiplicative group of a field must be cyclic.
Give a proof by example in the case when |G| = 100.

Prove that there is no positive integer n so that 25 < n2
< 36. Prove this by directly proving the negation. Your proof
must only use integers, inequalities and elementary logic. You may
use that inequalities are preserved by adding a number on both
sides, or by multiplying both sides by a positive number. You
cannot use the square root function. Do not write a proof by
contradiction.

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