Math 361, Spring 2019, Assignment 14

Read:

1. Section 33.

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 concerning existence, uniqueness, and dimension of splitting fields.
2. Theorem constraining the order of a finite field.
3. Existence theorem for finite fields.
4. Theorem concerning the roots of $x^{(p^n)}-x$ in any field of order $p^n$.
5. Theorem relating $GF(p^n)$ to the splitting field of $x^{(p^n)}-x\in\mathbb{Z}_p[x]$.
6. Uniqueness theorem for finite fields.
7. Primitive element theorem.

Solve the following problems:

1. Section 33, problems 1, 2, and 3.
2. Find an explicit isomorphism between the fields $F_1=\mathbb{Z}_2[x]/\left\langle x^3+x+1\right\rangle$ and $F_2=\mathbb{Z}_2[x]/\left\langle x^3+x^2+1\right\rangle$. (Hint: call the standard generators of these fields $\alpha$ and $\beta$ respectively. Any isomorphism $\phi:F_1\rightarrow F_2$ must take $\alpha$ to a root of the polynomial $x^3+x+1$ inside $F_2$.)

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