Math 361, Spring 2020, Assignment 14

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

1. $\mathrm{Cl}_E(F)$ (the relative algebraic closure of $F$ in $E$).
2. $\overline{\mathbb{Q}}$ (the field of algebraic numbers).
3. Algebraic number field.
4. Algebraically closed field.
5. Absolute algebraic closure (of a field $F$).
6. Splitting field (of a non-constant polynomial $f\in F[x]$).

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

1. Theorem relating algebraic elements to finite-dimensional subextensions.
2. Theorem concerning sums, products, opposites, and inverses of algebraic elements ("The algebraic elements of an extension $F\rightarrow E$ form a...")
3. Theorem giving five equivalent conditions, any of which can be used as the definition of the phrase "algebraically closed field."
4. Theorem concerning the existence and uniqueness of absolute algebraic closures.
5. Theorem concerning the existence and uniqueness of splitting fields.

Solve the following problems:

1. Prove that the real number $\sqrt[99]{5}+\sqrt[101]{7}+\sqrt{3}$ is algebraic over $\mathbb{Q}$. Can you find a rational-coefficient polynomial that annihilates it?
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 Lagrange-like corollary of the Dimension Formula is helpful.)

4. Construct a splitting field for the polynomial $x^4-x\in\mathbb{Z}_2[x]$.

5. (Optional) Construct a splitting field for the polynomial $x^8-x\in\mathbb{Z}_2[x]$. (This is a lengthy and non-trivial calculation, but if you do it you will cement many ideas introduced over the semester, and you will discover some strange and beautiful properties of finite fields.)

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