Splitting Fields: Construction, Quotients, and Examples

All flashcards

What is a splitting field?

An extension generated over the base field by a polynomial's roots, in which the polynomial factors completely into linear factors.

What does it mean for a polynomial to split?

It can be written as a nonzero constant times a product of linear factors over the field.

What is the splitting field of $X^2-2$ over $\mathbb{Q}$?

$\mathbb{Q}(\sqrt{2})$, which contains both roots $\sqrt{2}$ and $-\sqrt{2}$.

In what sense is a splitting field unique?

Any two splitting fields of the same polynomial over the same base field are isomorphic by an isomorphism fixing that field.

What is a normal extension in this context?

An extension that is a splitting field for a family of polynomials over its base field.

Quiz questions

1. What does it mean for a polynomial to split over a field?Answer: It factors completely into linear factors over that field.
2. Why is $K_i[X]/(f(X))$ a field when $f$ is irreducible over $K_i$?Answer: The ideal generated by $f$ is maximal in $K_i[X]$.
3. If $\alpha$ is the residue class of $X$ in $K_i[X]/(f(X))$, what equation does it satisfy?Answer: $f(\alpha)=0$
4. If $f$ is irreducible of degree $d$ over $K_i$, what is the degree of $K_i[X]/(f(X))$ over $K_i$?Answer: $d$
5. Which field is isomorphic to $\mathbb{R}[X]/(X^2+1)$?Answer: $\mathbb{C}$