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

• HardBoiled88
In summary, the conversation involves a student seeking help in catching up on their Algebra class while trying to understand field theory. They have been working on problems involving field K = Q[x]/(x^3 − 2) and field L = Z2[x]/(x3 + x + 1), and are asking for assistance in solving them. The problems involve finding the multiplicative inverse, polynomial multiplication and long division, and using linear algebra to find specific values. The conversation also includes a discussion on Lagrange's Theorem and factoring polynomials over Z3[x].
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 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].

HardBoiled88 said:
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 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.
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.

## What is field theory?

Field theory is a branch of physics that studies the interactions between objects or particles through the concept of fields. These fields are regions of space that have certain properties and can exert forces on objects within them.

## What types of problems can be solved using field theory?

Field theory can be applied to a wide range of physical phenomena, including electromagnetism, gravity, and quantum mechanics. It can also be used to study the behavior of particles and their interactions with each other and their environment.

## What are some common techniques for solving field theory problems?

Some common techniques for solving field theory problems include using mathematical equations and models, performing experimental measurements and observations, and using computer simulations to analyze complex systems.

## What are some real-world applications of field theory?

Field theory has many practical applications, such as in the development of electronics and technology, understanding the behavior of materials, and predicting and controlling natural phenomena like weather patterns and earthquakes.

## What are some resources for getting help with field theory problems?

There are many resources available for help with field theory problems, including textbooks, online tutorials and courses, physics forums and discussion boards, and working with a tutor or mentor. Your school or university may also have a physics department or tutoring center that can provide assistance.

• Linear and Abstract Algebra
Replies
6
Views
2K
• Special and General Relativity
Replies
1
Views
970
• Linear and Abstract Algebra
Replies
52
Views
2K
• Linear and Abstract Algebra
Replies
3
Views
2K
• Calculus and Beyond Homework Help
Replies
10
Views
953
• Quantum Physics
Replies
4
Views
1K
• General Math
Replies
4
Views
1K
• Linear and Abstract Algebra
Replies
2
Views
1K
• Linear and Abstract Algebra
Replies
10
Views
1K
• Linear and Abstract Algebra
Replies
3
Views
2K