Math 361, Spring 2015, Assignment 12

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

1. Additive representation (of $GF(p^n)$).
2. Multiplicative representation (of $GF(p^n)$).

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

1. Primitive element theorem.
2. Theorem concerning degrees of irreducible polynomials in $\mathbb{Z}_p[x]$.
3. Classification of subfields of $GF(p^n)$.
4. Theorem describing the algebraic closure of $\mathbb{Z}_p$ (this will necessarily be a somewhat imprecise statement, since we have not formally defined the term direct limit, and the meaning of "union" in this context is not quite clear).

Solve the following problems:

1. Section 33, problems 1, 3, 4, and 11.

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

Questions:

Does anybody have:

1. Primitive Element Theorem?
2. Classification of subfields of $GF(p^n)$?
3. Theorem concerning degrees of irreducible polynomials in $\mathbb{Z}_p[x]$?

Thanks.--Ian.Morse (talk) 20:52, 18 May 2015 (EDT)

1. There exists some $\alpha\in GF(p^n)$ such that $GF(p^n) = \{0, 1, \alpha, \alpha^2, \dots, \alpha^{p^n-2}\}$. Any such $\alpha$ is called a primitive element of $GF(p^n)$.
2. Any subfield of $GF(p^n)$ has order $p^d$ where $d$ is a divisor of $n$. Furthermore, for each $d$ dividing $n$, there is a unique subfield of order $p^d$.
3. There is at least one irreducible polynomial in each positive degree.

--Steven.Jackson (talk) 21:56, 18 May 2015 (EDT)