Math 361, Spring 2018, Assignment 8

Read:

1. Section 29.

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

1. Field extension.
2. Base field.
3. Extension field.
4. Injection (associated with a field extension).
5. Algebraic element (of an extension field).
6. Transcendental element (of an extension field).
7. Subextension.
8. Subextension generated by a subset.
9. Simple extension.
10. Finitely generated extension.
11. Minimal polynomial (of an algebraic element; a.k.a. $\mathrm{irr}(\beta,F)$).

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

1. Classification of simple extensions.
2. Theorem concerning irreducibility of minimal polynomials.

Solve the following problems:

1. Section 29, problems 1, 3, 5, 7, 9, 11, 13, and 15.
2. Consider the field $GF(4)=\mathbb{Z}_2[x]/\left\langle x^2+x+1\right\rangle$, and as usual let $\alpha$ denote the generator $x+\left\langle x^2+x+1\right\rangle$. Compute $\mathrm{irr}(\alpha,\mathbb{Z}_2)$. Then compute $\mathrm{irr}(\alpha+1,\mathbb{Z}_2)$.

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