Question

Let p(x) be an irreducible polynomial of degree n over a finite field K. Show that...

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.

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 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.
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 z0 be a zero of the polynomial P(z)=a0 +a1z+a2z2 +···+anzn of degree n (n ≥...
Let z0 be a zero of the polynomial P(z)=a0 +a1z+a2z2 +···+anzn of degree n (n ≥ 1). Show in the following way that P(z) = (z − z0)Q(z) where Q(z) is a polynomial of degree n − 1. (an ̸=0) (k=2,3,...). (a) Verify that zk−zk=(z−z)(zk−1+zk−2z +···+zzk−2+zk−1) 00000 (b) Use the factorization in part (a) to show that P(z) − P(z0) = (z − z0)Q(z) where Q(z) is a polynomial of degree n − 1, and deduce the desired result from...
Prove that for every positive integer n, there exists an irreducible polynomial of degree n in...
Prove that for every positive integer n, there exists an irreducible polynomial of degree n in Q[x].
Show the polynomial x4 − x3 − 2x2 + 6x − 4 is irreducible over Q.
Show the polynomial x4 − x3 − 2x2 + 6x − 4 is irreducible over Q.
a) Show that if K = ℤ2 then q = x3 + x + 1 ∈K...
a) Show that if K = ℤ2 then q = x3 + x + 1 ∈K [x] is irreducible. b) Prove: If p ∈ℝ [x] and degree of p = 3, then p is reducible. (Help: curve discussion, statement also applies if p has odd degree.)
Suppose n = rs where r and s are distinct primes, and let p be a...
Suppose n = rs where r and s are distinct primes, and let p be a prime. Determine (with proof, of course) the number of irreducible degree n monic polynomials in Fp[x]. (Hint: look at the proof for the number of prime degree polynomials) The notation Fp means the finite field with q elements
(Sage Exploration) In class, we primarily have worked with the field Q and its finite extensions....
(Sage Exploration) In class, we primarily have worked with the field Q and its finite extensions. For each p ∈ Z primes, we can also study the field Z/pZ , which I will also denote Fp, and its finite extensions. Sage understands this field as GF(p). (a) Define the polynomial ring S = F2[x]. (b) Find all degree 2 irreducible polynomials. How many are there? For each, completely describe the corresponding quadratic field extensions of F2. (c) True of false:...
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
(Modern Algebra) Show that every finite subgroup of the multiplicative group of a field is cyclical....
(Modern Algebra) Show that every finite subgroup of the multiplicative group of a field is cyclical. (Hint: consider m as the order of the finite subgroup and analyze the roots of the polynomial (x ^ m) - 1 in field F)
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT