Math 361, Spring 2018, Assignment 12

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

1. Splitting (of a polynomial).
2. Splitting field (of a polynomial).
3. Non-split part (of a polynomial).

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

1. Theorem concerning closure of the field of constructible numbers under square roots.
2. Theorem relating constructibility to the degree of the minimal polynomial.
3. Theorem concerning squaring the circle.
4. Theorem concerning duplication of the cube.
5. Theorem concerning the trisection of angles.

Solve the following problems:

1. Working over $\mathbb{Z}_2$, construct the splitting field of the polynomial $x^4-x$.
2. Let $F$ denote any field with eight elements (for example, you may take $F=\mathbb{Z}_2[x]/\left\langle x^3+x+1\right\rangle$). Factor the polynomial $x^8-x$ completely over $F$. (Hint: there are a lot of roots.)

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