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HardBoiled88

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Let field K = Q[x]/(x^3 − 2). (Assume x^3−2 is irreducible over Q.) All elements

should be written in the form a + bx + cx^2 with a, b, c^2 in Q.

(a) Find (x^2 + 3x + 7)(2x^2 + x + 3).

(b) Find x^4.

(c) Find x^30.

(d) Use the Extended Euclidean Algoritm to find the multiplicative inverse of x + 3.

(e) Write (alpha + beta*x + gamma*x^2)(x+3) in the form a+bx+c^2 where a, b, c are

explicit functions of alpha, beta, gamma. Now use linear algebra to find alpha, beta, gamma such that a = 1, b = 0, c = 0

another similar problem:

Let field L = Z2[x]/(x3 + x + 1). (Assume that x3 + x + 1 is irreducible over Z2.) All

elements should be written in the form a + bx + cx^2 with a, b, c^2 in Z2.

(a) Find (x^2 + x + 1)(x^2 + 1).

(b) Find x^4.

(c) Find x^70.

(d) Use the Extended Euclidean Algoritm to find the multiplicative inverse of x + 1

(e) Write (alpha+beta*x+gamma*x^2

)(x + 1) in the form a + bx + c^2 where a, b, c

are explicit functions of alpha, beta, gamma

. Now use linear algebra to find alpha, beta, gamma

such that a = 1, b = 0, c = 0.

Let p be prime, let f(x) in Zp[x] be irreducible with degree d and set K = Zp[x]/(f(x)). K is then a field. Let K* be the multiplicative group of all nonzero elements of K.

(a) How many elements does K have?

(b) How many elements does K* have?

(c) Use Lagrange’s Theorem to prove that:

x^(p^(d)-1) = 1 in K

(d) Deduce that:

f(x) | [x^(p^(d)-1) − 1] in Zp[x]

(e) Use the above to factor x^26 − 1 into irreducible polynomials over Z3[x].