Question

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.

Answer #1

Let p(x) be an irreducible polynomial of degree n over a finite
field K. Show that its Galois group over K is cyclic of order n and
then show how the Galois group of x3 − 1 over Q is
cyclic of order 2.

Let f(x) be a cubic polynomial of the form x^3 +ax^2 +bx+c with
real coefficients.
1. Deduce that either f(x) factors in R[x] as the product of
three degree-one
polynomials, or f(x) factors in R[x] as the product of a
degree-one
polynomial and an irreducible degree-two polynomial.
2.Deduce that either f(x) has three real roots (counting
multiplicities) or
f(x) has one real root and two non-real (complex) roots that are
complex
conjugates of each other.

True or False, explain:
1. Any polynomial f in Q[x] with deg(f)=3 and no roots in Q is
irreducible.
2. Any polynomial f in Q[x] with deg(f)-4 and no roots in Q is
irreducible.
3. Zx40 is isomorphic to
Zx5 x Zx8
4. If G is a finite group and H<G, then [G:H] = |G||H|
5. If [G:H]=2, then H is normal in G.
6. If G is a finite group and G<S28, then there is
a subgroup of G...

Let p be a prime and m an integer. Suppose that the polynomial
f(x) = x^4+mx+p is reducible over Q. Show that if f(x) has no zeros
in Q, then p = 3.

Let f ∈ Z[x] be a nonconstant polynomial. Prove that the set S =
{p prime: there exist infinitely many positive integers n such that
p | f(n)} is infinite.

show that any polynomial of odd degree has at least one real
root

Let b be a primitive root for the odd prime p. Prove that b^k is
a primitive root for p if and only if gcd(k, p − 1) = 1.

Let F be a field and let f(x) be
an element of F[x] be an an irreducible
polynomial. Suppose K is an extension field containing F and that
alpha is a root of f(x). Define a function f: F[x] ---> K by
f:g(x) = g(alpha). Prove the ker(f) =<f(x)>.

Let
a1, a2, ..., an be distinct n (≥ 2) integers. Consider the
polynomial
f(x) = (x−a1)(x−a2)···(x−an)−1 in Q[x]
(1) Prove that if then f(x) = g(x)h(x)
for some g(x), h(x) ∈ Z[x],
g(ai) + h(ai) = 0 for all i = 1, 2, ..., n
(2) Prove that f(x) is irreducible over Q

Let p be an odd prime, and let x = [(p−1)/2]!. Prove that x^2 ≡
(−1)^(p+1)/2 (mod p).
(You will need Wilson’s theorem, (p−1)! ≡−1 (mod p).) This gives
another proof that if p ≡ 1 (mod 4), then x^2 ≡ −1 (mod p) has a
solution.

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