Math 361, Spring 2014, Assignment 3

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

1. Monic polynomial.
2. Monic generator (of a principal ideal).
3. Field extension.
4. Extension field.
5. Base field.
6. Injection (associated with a field extension).

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

1. Classification of prime ideals in $F[x]$.
2. Kronecker's Theorem.

Solve the following problems:

1. Consider the polynomial $f = x^3 - 2$ in $\mathbb{Q}[x]$.
   • Show that $f$ is irreducible over $\mathbb[Q]$.
   • Now consider the extension field $E_1 = \mathbb{Q}[x]/\langle f\rangle$. Is $f$ still irreducible when regarded as a polynomial with coefficients in $E_1$? If not, then write an explicit non-trivial factorization of $f$ over $E_1$.
   • Does $f$ split over $E_1$? (Warning: this is a hard question. In principle you can answer it now, with a lot of work, but soon we will learn an efficient way to answer it. If you prefer, take a guess at this and forge ahead.) If not, can you produce another extension $E_2$ over which it does split?

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