Question

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

Answer #1

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

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 p be an odd prime. Let f(x) ∈ Q(x) be an irreducible
polynomial of degree p whose Galois group is the dihedral group
D_2p of a regular p-gon. Prove that f (x) has either all real roots
or precisely one real root.

Prove or disprove that 3|(n^3 − n) for every positive integer
n.

Definition of Even: An integer n ∈ Z is even if there exists an
integer q ∈ Z such that n = 2q.
Definition of Odd: An integer n ∈ Z is odd if there exists an
integer q ∈ Z such that n = 2q + 1.
Use these definitions to prove the following:
Prove that zero is not odd. (Proof by contradiction)

a)Give an example of a polynomial with integer coefficients of
degree at least 3 that has at least 3 terms that satisfies the
hypotheses of Eisenstein's Criterion, and is therefore
irreducible.
b)Give an example of a polynomial with degree 3 that has at
least 3 terms that does not satisfy the hypotheses of Eisenstein's
Criterion.

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 the set V of all polynomials of degree ≤ n including
the zero polynomial is vector space over the field R under usual
polynomial addition and scalar multiplication. Further, find the
basis for the space of polynomial p(x) of degree ≤ 3. Find a basis
for the subspace with p(1) = 0.

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

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 45 minutes ago

asked 49 minutes ago

asked 51 minutes ago

asked 55 minutes ago

asked 56 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago