Math 361, Spring 2016, Assignment 12

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

1. Polynomial with distinct roots.
2. Formal derivative.
3. $GF(q)$.
4. Solvable by radicals (an example suffices; good luck producing a non-example at this stage).
5. Symmetry (of a field extension $F\rightarrow E$).
6. Galois group (of a field extension).

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

1. Theorem concerning elementary properties of the formal derivative.
2. Theorem relating formal derivatives to repeated roots.
3. Theorem concerning the existence of $GF(q)$.
4. Primitive element theorem.
5. Theorem concerning the distribution of irreducibles in $\mathbb{Z}_p[x]$.

Solve the following problems:

1. Section 51, problem 15(b-c).
2. Calculate the Galois group of $\mathbb{Q}\rightarrow\mathbb{Q}(\sqrt{2})$. (Hint: imitate our calculation of the Galois group of $\mathbb{R}\rightarrow\mathbb{C}$. Where can a symmetry send $\sqrt{2}$?)
3. Calculate the Galois group of $\mathbb{Q}\rightarrow\mathbb{Q}(\sqrt[3]{2})$. (Hint: where do the roots of $x^3-2$ lie in the complex plane?)
4. (Challenge) Calculate the Galois group of the splitting field of $x^3-2$ over $\mathbb{Q}$. (The answer is definitely different from the previous problem.)

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