Math 361, Spring 2014, Assignment 2

Carefully define the following terms, then give one example and one non-example of each:

1. Maximal ideal.
2. Prime ideal.
3. Prime subfield (of a given field).

Carefully state the following theorems (you need not prove them):

1. Theorem relating maximal ideals to fields.
2. Theorem relating prime ideals to integral domains.
3. Theorem relating maximal ideals to prime ideals.
4. Classification of ideals in $F[x]$ (where $F$ is a field).
5. Classification of maximal ideals in $F[x]$.

Solve the following problems:

1. Section 27, problems 3, 5, 16, 18, and 19.
2. Construct a field with exactly four elements. Give names to the elements, then write addition and multiplication tables. Verify that your structure is a field by identifying the multiplicative inverse of each non-zero element. (Hint: begin by finding a quadratic irreducible polynomial in $\mathbb{Z}_2[x]$.)
3. Describe (but do not carry out) constructions producing fields of order 8, 9, 16, 25, and 27. For which integers $n$ do you think your methods can produce fields of order $n$?

--------------------End of assignment--------------------

Solutions:

Definitions:

1. Maximal Ideal

   Given a ring \(R\), an ideal \(I\) is maximal if \(I \neq R\) and there is no ideal \(J \neq R\) such that \(I \subset J\). So you can\t make \(I\) bigger without including the whole ring.

   Example:

   \(5\mathbb{Z}\) is a maximal ideal in \(\mathbb{Z}\). This is because 5 has no factors.

   Non-example:

2. Prime Ideal

   An ideal \(I\) is prime if \(ab \in I\) implies \(a\in I\) or \(b\in I\).

   Example:

   \(5\mathbb{Z}\) is also a prime ideal.

   Non-example:

3. Prime Subfield

   Given a field \(F\), there is a subfield of \(F\) isomorphic to \(\mathbb{Q}\) or \(\mathbb{Z}_p\).

   Example:

   \(\mathbb{Q}\) is a prime subfield of \(\mathbb{R}\).

   Non-example:

Theorems:

1. Theorem Relating Maximal Ideals to Fields

   Given a ring \(R\) and an ideal \(I\) of \(R\), \(R/I\) is a field if and only if \(I\) is maximal.

2. Theorem Relating Prime Ideals to Integral Domains

   Given a ring \(R\) and an ideal \(I\), \(R/I\) is an integral domain if and only if \(I\) is prime.

3. Theorem Relating Maximal Ideals to Prime Ideals

   All maximal ideals are prime ideals.

4. Classification of Ideals in \(F[x]\)

   All ideals of \(F[x]\) are principal, meaning they are generated by a single element.

5. Classification of Maximal Ideals in \(F[x]\).

   An ideal of \(F[x]\) is maximal if and only if it is generated by an irreducible polynomial.

Exercises:

1. 2

   Quadratic polynomial: \(p = x^2 + x + 1\). This is irreducible: \(p(0) = 0 +0 + 1 =1\), \(p(1) = 1 + 1 +1 =1\). Since this is quadratic, the only elements of \(\mathbb{Z}_2/\langle p \rangle\) are linear and constant polynomials, i.e. \(0,1,x,x+1\). Addition:

$$ \begin{array}{c|c|c|c|c} & 0 & 1 & x & x+1\\ 0 & 0 & x & x & x+1\\ 1 & 1 & 0 & x+1 & x\\ x & x & x+1 & 0 & 1\\ x+1 & x+1 & x & 1 & 0 \end{array} $$

Multiplication: $$ \begin{array}{c|c|c|c|c} & 0 & 1 & x & x+1\\ 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 1 & x & x+1\\ x & 0 & x & x+1 & 1\\ x+1 & 0 & x+1 & 1 & x \end{array} $$

1. 3

   If we start with \(\mathbb{Z}_p\), and we mod out by the ideal generated by an irreducible polynomial of order \(n\), \we get a new field containing all the polynomials up to degree \(n-1\). So we get \(p\) constant polynomials, \(p^2\) linear polynomials, and in general we get \(p^{k+1}\) polynomials of degree \(k\).