How Can I Catch Up on Field Theory After Being Sick?

[HardBoiled88]

I've been sick with mono for the past month and am trying to catch up in my Algebra class, but being so far behind I'm having a lot of trouble trying to grasp so much in so little time. Currently, I'm trying to get my head around field theory. Here are are few problems I've been working on. Is someone kind enough to walk me through them. Thank you for any help you can give me.

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 algorithm 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 algorithm 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].

 

[fresh_42]

I've been sick with mono for the past month and am trying to catch up in my Algebra class, but being so far behind I'm having a lot of trouble trying to grasp so much in so little time. Currently, I'm trying to get my head around field theory. Here are are few problems I've been working on. Is someone kind enough to walk me through them. Thank you for any help you can give me.

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 algorithm 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 algorithm 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.

This is polynomial multiplication followed by long division, which is done this way:
https://www.physicsforums.com/threa...r-a-polynomial-over-z-z3.889140/#post-5595083

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].

a) Which maximal power occurs in a polynomial of $\mathbb{Z}_p[x]/(f(x))$? Now how many possibilities are there for each coefficient?
b) $K$ is an integral domain, since $f$ is irreducible. So how many zero divisors are there in an integral domain?
The rest is as suggested, and e) again a long division.