Question

Let p be an odd prime. Let f(x) ∈ Q(x) be an irreducible polynomial of degree...

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.

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let f(x) be a cubic polynomial of the form x^3 +ax^2 +bx+c with real coefficients. 1....
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...
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...
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:...
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
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...
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 a1, a2, ..., an be distinct n (≥ 2) integers. Consider the polynomial 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...
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.
The polynomial of degree 4, P(x) has a root of multiplicity 2 at x=2 and roots...
The polynomial of degree 4, P(x) has a root of multiplicity 2 at x=2 and roots of multiplicity 1 at x=0 and x=-4 It goes through the point (5,324). Find a formula for P(x)
Use the intermediate value theorem (only) to prove that any real polynomial of odd degree has...
Use the intermediate value theorem (only) to prove that any real polynomial of odd degree has at least one real solution. Is the same conclusion true if the degree is even?
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT