Abstract Algebra: Theory and Applications · Problem 21.5.1

Finite Fields — Thomas Judson

Mathematics — Abstract Algebra · algebraic-numbers-minimal-polynomial

Problem

Show that each of the following numbers is algebraic over

\mathbb{Q}

by finding the minimal polynomial of the number over

\mathbb{Q}

. (a)

\sqrt{1/3 + \sqrt{7}}

(b)

\sqrt{3} + \sqrt[3]{5}

(c)

\sqrt{3} + \sqrt{2}\, i

(d)

\cos\theta + i\sin\theta

for

\theta = 2\pi/n

with

n \in \mathbb{N}

(e)

\sqrt{\sqrt[3]{2} - i}

Hint. (a)

x^4 - (2/3)x^2 - 62/9

; (c)

x^4 - 2x^2 + 25