Question

Consider the polynomial f(x) = x ^4 + x ^3 + x ^2 + x +...

Consider the polynomial f(x) = x ^4 + x ^3 + x ^2 + x + 1 with roots in GF(256). Let b be a root of f(x), i.e., f(b) = 0.

The other roots are b^ 2 , b^4 , b^8 .

e) Write b 4 as a combination of smaller powers of b.

Prove that b 5 = 1. f) Given that b 5 = 1 and the factorization of 255, determine r such that b = α r

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.
Given the polynomial f(x) = 3x^4 -7x^3 -3x^2 + 17x + 10, and given the zeros...
Given the polynomial f(x) = 3x^4 -7x^3 -3x^2 + 17x + 10, and given the zeros of -1 and -2/3,   A). Write the polynomial in linear and irreducible quadratic factors B). Write the linear factorization of the polynomial.
3. (50) Let f(x) = x^4 + 2. Find a factorization of f(x) into irreducible polynomials...
3. (50) Let f(x) = x^4 + 2. Find a factorization of f(x) into irreducible polynomials in each of the following rings, justifying your answers briefly: (i) Z3 [x]; (ii) Q[x] (this can be done easily using an appropriate theorem); (iii) R[x] (hints: you may find it helpful to write γ = 2^(1/4), the positive real fourth root of 2, and to consider factors of the form x^2 + a*x + 2^(1/2); (iv) C[x] (you may leave your answer in...
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...
The polynomial f(x)= 2(x-4)2(x+5)3. Give the y intercept, roots with multiplicities, and describe end behavior
The polynomial f(x)= 2(x-4)2(x+5)3. Give the y intercept, roots with multiplicities, and describe end behavior
uppose a is a simple root of the polynomial f(x) and g(x) is another polynomial of...
uppose a is a simple root of the polynomial f(x) and g(x) is another polynomial of degree > degree of f(x). Then g(x)/f(x) = (A/x-a) + other terms. Prove that A= g(a)/f'(a) by not using L'Habitial's rule.
For each polynomial f(x) ∈ Z[x], let f ' (x) denote its derivative, which is also...
For each polynomial f(x) ∈ Z[x], let f ' (x) denote its derivative, which is also a polynomial in Z[x]. Let R be the following subset of Z[x]: R = {f(x) ∈ Z[x] | f ' (0) = 0}. (a) Prove that R is a subring of Z[x]. (b) Prove that R is not an ideal of Z[x].
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
1. a) For the polynomial f(x) = ?4 − 4?^3 + 22?^2 + 28?− 203, find...
1. a) For the polynomial f(x) = ?4 − 4?^3 + 22?^2 + 28?− 203, find the following: a. Find all the zeros using the given zero ? = 2 − 5?. Write the zeros in exact form. b. Factor f(x) as a product of linear factors. Zeros: x = x = x= x=
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT