Help with field theory problems

In summary, the conversation discusses the struggles of a person who has been sick and is trying to catch up in their Algebra class. They are currently working on field theory problems and have requested help in solving them. The problems involve finding elements and using the Extended Euclidean Algorithm to find multiplicative inverses. The conversation also mentions using linear algebra and Lagrange's Theorem to solve for specific values. Finally, an example problem involving irreducible polynomials over Z3[x] is given.
  • #1
HardBoiled88
6
0
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].
 
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  • #2
What have you tried so far?
 

Related to Help with field theory problems

1. What is field theory and how does it relate to physics?

Field theory is a branch of physics that describes how objects interact with each other through fields. These fields can be thought of as invisible forces that permeate space and influence the behavior of particles and objects within them. In physics, field theory is used to explain phenomena such as electromagnetism, gravity, and quantum mechanics.

2. What are some common problems encountered in field theory?

Some common problems encountered in field theory include calculating the behavior of particles in a given field, understanding the effects of multiple fields on a single particle, and determining the properties and interactions of different types of fields.

3. How can I approach solving field theory problems?

One approach to solving field theory problems is to start by carefully defining the problem and identifying the relevant fields and particles involved. From there, you can use mathematical equations and principles to analyze and calculate the behavior of the system. It can also be helpful to break the problem down into smaller, more manageable parts and consider any simplifying assumptions that can be made.

4. Are there any common mistakes to avoid when working on field theory problems?

Some common mistakes to avoid when working on field theory problems include using incorrect equations or assumptions, not considering all relevant factors and interactions, and not double-checking your calculations. It can also be helpful to carefully read and interpret the problem statement to ensure you are addressing the intended question.

5. How can I improve my understanding and skills in field theory?

To improve your understanding and skills in field theory, it can be helpful to practice solving a variety of problems and working through examples with a tutor or study group. It can also be beneficial to read and study textbooks and articles on the subject, and to stay current with advancements and research in the field. Experimentation and hands-on experience with field theory concepts can also deepen your understanding and knowledge.

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