Abstract algebra questions relating to Ideals and cardinality of factor rings

In summary, the task is to find the number of elements in the ring Z_5[x]/I, where I is generated by x^4+4 and x^2+3x+1. The student has attempted to find the zeros of the generating polynomial for part a, but is unsure if it helps. The problem is from a practice test, but the concept has not been covered in class or referenced in the textbook. The question also asks for a description or list of elements in the ring.
  • #1
cloverforce
3
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.
 
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  • #2
Can you describe or list the elements in [tex]\mathbb{Z}_5[x]/I[/tex]?
 

Related to Abstract algebra questions relating to Ideals and cardinality of factor rings

1. What are ideals in abstract algebra?

Ideals are subsets of a ring that have certain properties such as closure under addition and multiplication by elements in the ring. They can be thought of as generalizations of the concept of divisibility in integers.

2. How are ideals related to factor rings?

Factor rings are created by "factoring out" an ideal from a ring. This means that the elements of the ideal are considered equivalent to 0 in the factor ring, and the operations are defined accordingly. For example, in the ring of integers, the ideal generated by 2 corresponds to the factor ring of even integers.

3. What is the relationship between ideals and subrings?

Ideals are a special type of subring, where the subset also has the property of absorbing the elements of the ring. In other words, multiplying an element in the ring by an element in the ideal will result in an element in the ideal.

4. How does the cardinality of factor rings relate to the cardinality of the original ring?

The cardinality of a factor ring is equal to the cardinality of the original ring divided by the cardinality of the ideal being factored out. This can be seen intuitively as "grouping together" elements that are considered equivalent in the factor ring.

5. Can ideals only be generated by one element?

No, ideals can be generated by any number of elements in a ring. These are known as finitely or infinitely generated ideals. In some cases, a ring may have ideals that cannot be generated by any finite number of elements.

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