Question

Let F be a ** field** and let f(x) be

Answer #1

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
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.)

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.

Let F be an ordered field. Let S be the subset [a,b)
i.e, {x|a<=x<b, x element of F}. Prove that infimum and
supremum exist or do not exist.

Let
F be a field and let a(x), b(x) be polynomials in F[x]. Let S be
the set of all linear combinations of a(x) and b(x). Let d(x) be
the monic polynomial of smallest degree in S. Prove that d(x)
divides a(x).

True/False, explain:
1. If G is a finite group and G28, then there is a subgroup of G of
order 2401=74
2. If |G|=19, then G is isomorphic to Z19.
3. If F subset of K is a degree 5 field extension, any element b in
K is the root of some polynomial p(x) in F[x]
4. If F subset of K is a degree 5 field extension, viewing K as
a vector space over F, Aut(K, F) consists of...

Let F be a field and f(x), g(x) ? F[x] both be of degree ? n.
Suppose that there are distinct elements c0, c1, c2, · · · , cn ? F
such that f(ci) = g(ci) for each i. Prove that f(x) = g(x) in
F[x].

4. Let f : G→H be a group homomorphism. Suppose a∈G is an
element of finite order n.
(a) Prove that f(a) has finite order k, where k is a divisor of
n.
(b) If f is an isomorphism, prove that k=n.

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
f(x) be polynomial function in field F[x].
f’(x) be the derivative of f(x).
Given the greatest common factor (f(x), f’(x))=1.
And (x-a)|f(x). Show that (x-a)^2 can not divide f(x).

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 5 minutes ago

asked 33 minutes ago

asked 33 minutes ago

asked 33 minutes ago

asked 42 minutes ago

asked 48 minutes ago

asked 52 minutes ago

asked 59 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago