Math 361, Spring 2016, Assignment 9

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

1. (Relative) algebraic closure (of $F$ in $E$).
2. Field of algebraic numbers.
3. Algebraic number field.
4. Algebraically closed field.
5. (Absolute) algebraic closure (of $F$).
6. Discriminant (of a quadratic polynomial).
7. Construction by compass and straightedge.

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

1. Theorem concerning sums, products, and inverses of algebraic elements of a field extension.
2. Theorem giving five equivalent conditions for a field to be algebraically closed.
3. Theorem concerning existence and uniqueness of (absolute) algebraic closures.
4. Quadratic formula.
5. Theorem bounding the dimension of the subextension generated by the roots of a quadratic polynomial.

Solve the following problems:

1. Section 31, problems 1, 3, 5, 7, 18, 23, and 24.
2. Show that $\overline{\mathbb{Q}}$ is algebraically closed, as follows: let $p=a_0+\dots+a_nx^n$ be any non-constant polynomial with coefficients in $\overline{\mathbb{Q}}$.

(a) Explain why $p$ must have a root in $\mathbb{C}$. Choose such a root, and call it $\beta$.

(b) Using the Dimension Theorem, show that $\mathbb{Q}(a_0,\dots,a_n)$ (i.e. the smallest subfield containing $\mathbb{Q}$ and the coefficients of $p$) must be finite-dimensional over $\mathbb{Q}$.

(c) Using the classification of simple extensions, show that $\mathbb{Q}(a_0,\dots,a_n,\beta)$ is finite-dimensional over $\mathbb{Q}(a_0,\dots,a_n)$.

(d) Using the Dimension Theorem a second time, conclude that $\beta$ is algebraic over $\mathbb{Q}$, and hence lies in $\overline{\mathbb{Q}}$.

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