Math 361, Spring 2014, Assignment 14

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

1. Splitting field (of a nonconstant polynomial $p\in F[x]$).
2. Galois group (of a nonconstant $p\in F[x]$).

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

1. Theorem on existence and uniqueness of splitting fields.
2. Theorem concerning finiteness of the Galois group.

Solve the following problems:

1. Give an example of a domain which is not a UFD.
2. Consider the polynomials $p=x^2-2, q=x^3-2,$ and $r=x^4-2$ in $\mathbb{Q}[x]$. Let $E_1, E_2,$ and $E_3$ be the splitting fields of $p, q,$ and $r$, respectively, over $\mathbb{Q}$. Compute the dimensions $[E_1:\mathbb{Q}], [E_2:\mathbb{Q}],$ and $[E_3:\mathbb{Q}]$. (Hint: follow the construction used to prove existence of splitting fields, and use the Dimension Formula. To compute the dimension of $E_3$, you may find the following fact helpful: $x^4-2 = (x^2 + \sqrt{2})(x^2-\sqrt{2}) = (x^2 + \sqrt{2})(x + \sqrt[4]{2})(x - \sqrt[4]{2})$.)
3. Compute the Galois group of $x^2+1\in\mathbb{R}[x]$.

Questions:

1. I just want to make sure my definition for Galois Group is right. When it is a non-constant polynomial and a field, I believe it is the splitting field of that non-constant polynomial over the field. But there was also an important Galois Group definition with two splitting fields where the group contains all isomophisms making the commutative diagram.

Solutions:

Definition: Splitting field (of a nonconstant polynomial $p\in F[x]$).
Fix a polynomial of strictly positive degree $p \in F[x]$. A splitting field for $p$ is a field extension $F \to E$ for which both of the following are true:

1. p splits over E
2. $E=F(\alpha_1,\alpha_2,\dots,\alpha_n)$,and every $\alpha_i$ is a root of $p$.

Example: Working over $\mathbb{R}$, $\mathbb{C}$ is a splitting field for $x^2+1$.

Definition: Galois Group (of a nonconstant $p\in F[x]$).
Fix a polynomial of strictly positive degree $p \in F[x]$. Define the Galois Group of $p$, denoted $Gal_F(p)$, to be the splitting field of $p$ over $F$.
Example: $Gal_{\mathbb{R}}(x^2+1)=\mathbb{C}$