Abstract algebra questions relating to Ideals and cardinality of factor rings

Click For Summary
SUMMARY

The discussion focuses on determining the number of elements in the ring Z_5[x]/I, specifically for two ideals: a) the ideal generated by the polynomial x^4 + 4, and b) the ideal generated by both x^4 + 4 and x^2 + 3x + 1. For part a, the zeros of the polynomial in Z_5 are identified as 1, 2, 3, and 4, which indicates that the ideal has a finite number of elements. The solution requires understanding the structure of factor rings and the implications of polynomial degrees in finite fields.

PREREQUISITES
  • Understanding of ring theory and factor rings
  • Familiarity with polynomial rings over finite fields, specifically Z_5[x]
  • Knowledge of ideals and their generation in algebraic structures
  • Basic concepts of cardinality in set theory
NEXT STEPS
  • Study the structure of finite fields and their polynomial rings
  • Learn about the properties of ideals in ring theory
  • Explore the concept of cardinality in algebraic structures
  • Investigate the application of the Chinese Remainder Theorem in polynomial factorization
USEFUL FOR

Students of abstract algebra, particularly those preparing for exams involving ring theory, polynomial rings, and ideals. This discussion is beneficial for anyone looking to deepen their understanding of factor rings and their cardinality.

cloverforce
Messages
3
Reaction score
0

Homework Statement


Find the number of elements in the ring Z_5[x]/I, where I is a) the ideal generated by x^4+4, and b) where I is the ideal generated by x^4+4 and x^2+3x+1.


Homework Equations


Can't think of any.


The Attempt at a Solution


I started by finding the zeros of the generating polynomial for part a (which are 1, 2, 3, and 4 in Z_5), but I'm not even sure if that helps. This problem is from a list of practice problems for a test, but they're all of a type which we haven't covered in class, and I can't find any reference to anything like this in my textbook.
 
Physics news on Phys.org
Can you describe or list the elements in \mathbb{Z}_5[x]/I?
 

Similar threads

Principle Ideals of a Polynomial Quotient Ring
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Definition of minimal ideal using symbolic logic notation
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Localizing single variable quotient polynomial ring at a prime ideal
  • · Replies 6 ·
Replies
6
Views
2K
Abstract Algebra: Ring Proof (Multiplicative Inverse)
  • · Replies 1 ·
Replies
1
Views
2K
Abstract Algebra: Another Ring Proof
  • · Replies 9 ·
Replies
9
Views
2K
Undergrad Determining isomorphism for ##\frac{R}{(a, b)}##
  • · Replies 4 ·
Replies
4
Views
2K
Abstract Algebra: Rings, Unit Elements, Fields
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Meaning of "passage to the quotient"?
  • · Replies 3 ·
Replies
3
Views
1K
Abstract Algebra Homework Solution - Check Ring Homomorphism
  • · Replies 5 ·
Replies
5
Views
2K
Factoring Polynomials [Abstract Algebra]
  • · Replies 17 ·
Replies
17
Views
2K