Math 361, Spring 2021, Assignment 13

Read:

1. Section 26.
2. Section 27.

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

1. Maximal ideal.
2. Principal ideal domain.
3. Prime ideal (we did not discuss this in class; it is Definition 27.13 in the text).

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

1. Lemma regarding ideals which contain units.
2. Theorem relating maximal ideals to fields.
3. Containment criterion for principal ideals.
4. Equality criterion for principal ideals.
5. Theorem concerning ideals of $\mathbb{Z}$ (i.e. "$\mathbb{Z}$ is a...").
6. Theorem concerning ideals of $F[x]$ (i.e. "$F[x]$ is a...").
7. Theorem relating maximal ideals to irreducible elements in principal ideal domains.
8. Theorem relating prime ideals to integral domains (we did not discuss this in class; it is Theorem 27.15 in the text).
9. Theorem relating maximal ideals to prime ideals (we did not discuss this in class; it is Corollary 27.16 in the text).

Solve the following problems:

1. Section 26, problem 24. (Hint: Since $F$ is a field and is isomorphic to $F/\{0\}$, we know that $\{0\}$ is a maximal ideal; from this we can write a list of all the ideals of $F$.)
2. Section 27, problems 1, 2, 3, 5, 7, 9, and 18.
3. Find an irreducible polynomial of degree $4$ in $\mathbb{Z}_2[x]$. (Hint: this can be done with the Sieve, but there is a shorter way. First make a list of all irreducible polynomials of degree $2$; these are just the quadratics with no roots, and there is a simple pattern that predicts which polynomials will have roots in $\mathbb{Z}_2$. Next, write down a degree $4$ polynomial which has no roots, and test it for divisibility by the irreducible quadratics. If your root-free quartic is divisible by an irreducible quadratic, throw it away and try another root-free quartic, continuing until you find a root-free quartic which is not divisible by any irreducible quadratic.)
4. Construct a field with exactly sixteen elements. (Hint: the result of the previous question is directly relevant to this one.) Prove that your ring really is a field, without writing down the whole multiplication table.

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