Abstract Algebra: Theory and Applications · Problem 21.5.3

Finite Fields — Thomas Judson

Mathematics — Abstract Algebra · splitting-fields

Problem

Find the splitting field for each of the following polynomials. (a)

x^4 - 10x^2 + 21

over

\mathbb{Q}

(b)

x^4 + 1

over

\mathbb{Q}

(c)

x^3 + 2x + 2

over

\mathbb{Z}_3

(d)

x^3 - 3

over

\mathbb{Q}

Hint. (a)

\mathbb{Q}(\sqrt{3}, \sqrt{7})