Math 361, Spring 2019, Assignment 13

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

1. Relative algebraic closure (of $F$ inside $E$).
2. Algebraic extension.
3. Algebraically closed field.
4. (Absolute) algebraic closure (of a field $F$).
5. Splitting (of a polynomial).
6. Splitting field (of a polynomial).

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

1. Lagrange-like corollary of the Dimension Formula.
2. Theorem concerning sums, products, and inverses of algebraic elements ("The algebraic elements of $E$ form a...")
3. Theorem giving several conditions equivalent to the condition that $F$ is algebraically closed.
4. Theorem concerning existence and uniqueness of algebraic closures.
5. Fundamental Theorem of Algebra.

Solve the following problems:

1. Section 31, problems 24 and 29.
2. Recall that $\overline{\mathbb{Q}}$ denotes field of algebraic numbers, i.e. the relative algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$. Prove that $\overline{\mathbb{Q}}$ is algebraically closed, as follows:

(a) Suppose $p\in\overline{\mathbb{Q}}[x]$ is not constant. Explain why $p$ must have some root $\alpha$ in $\mathbb{C}$.

(b) Explain why each coefficient of $p$ must lie in some finite-dimensional subextension of $\mathbb{Q}\rightarrow\mathbb{C}$.

(c) Using the Dimension Formula, show that all of the coefficients of $p$ lie in one big finite-dimensional subextension (say $K$) of $\mathbb{Q}\rightarrow\mathbb{C}$.

(d) Explain why $\alpha$ must lie in some finite-dimensional subextension (say $E$) of $K\rightarrow\mathbb{C}$.

(e) Using the Dimension Formula again, show that $E$ is finite-dimensional over $\mathbb{Q}$.

(f) Conclude that $\alpha$ is algebraic over $\mathbb{Q}$, and hence lies in $\overline{\mathbb{Q}}$.

(Note that this argument is not really specific to $\overline{\mathbb{Q}}$; in fact the same argument proves that, whenever $F\rightarrow E$ is an extension and $E$ is algebraically closed, then the relative algebraic closure of $F$ in $E$ is also algebraically closed.)

3. Construct a splitting field for the polynomial $x^5-3x^3-2x^2+6\in\mathbb{Q}[x]$. (Hint: although this polynomial does not split over $\mathbb{Q}$, it does have a nontrivial factorization over $\mathbb{Q}$. Begin by using the high school technique of factoring by grouping. At a later stage of the problem, the result of Exercise 24 is also helpful.)

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