Math 361, Spring 2015, Assignment 11

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

1. $GF(p^n)$.

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

1. Theorem constraining the order of a finite field.
2. Uniqueness theorem for finite fields.
3. Existence theorem for finite fields.
4. The freshman's dream.

Solve the following problems:

1. Utilize the library or internet resources to learn about the Sieve of Eratosthenes, which is an ancient and simple method for producing lists of prime numbers. Then adapt the Sieve to produce lists of irreducible polynomials over $\mathbb{Z}_p$.
2. Find all irreducible cubic polynomials over $\mathbb{Z}_2$.
3. Construct at least two concrete models for $GF(8)$. Is it obvious that these two fields are isomorphic? Can you construct an explicit isomorphism? (Be persistent!) How many isomorphisms can you construct?

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

Solutions:

Problems:

3. Let $F_1=\mathbb{Z}_2/<x^3+x+1>$ and let $F_2=\mathbb{Z}_2/<x^3+x^2+1>$. Denote $\alpha=x+<x^3+x+1> \in F_1$ and $\beta=x+<x^3+x^2+1> \in F_2$. Then the map $\phi: F_1 \rightarrow F_2$ defined by $\phi(1_{F_1})=1_{F_2}$ and $\phi(\alpha)=\beta^2$ is an isomorphism.

This map can't be a well-defined homomorphism, since $\phi(0)=\phi(\alpha^3+\alpha+1)=\beta^6+\beta^2+1=\beta+1\neq0$. You need to send $\alpha$ to some element of $F_2$ that is killed by the minimal polynomial of $\alpha$. This is the part that takes persistence. There are exactly three possibilities, which one finds by trial and error: $\beta^2+1$ or $\beta^2+\beta$ or $\beta^2+\beta+1$. Choosing one of these, you can use the general form of the Fundamental Theorem of Homomorphisms (an anachronism for this assignment, I know) to build a well-defined isomorphism from $F_1$ to $F_2$. -Steven.Jackson (talk) 16:48, 1 May 2015 (EDT)