I am reading Nicholson: Introduction to Abstract Algebra, Section 6.2 Algebraic Extensions.

Example 13 on page 282 (see attachment) reads as follows:

"If $\displaystyle u = \sqrt[3]{2} $ show that $\displaystyle \mathbb{Q}(u) = \mathbb{Q}(u^2) $"

In the third line of the explanation - see page 282 of attachment - we read:

"But $\displaystyle [\mathbb{Q}(u^2) \ : \ \mathbb{Q}] \ne 1 $ because $\displaystyle u^2 \notin \mathbb{Q} $ ... ... "

Can someone explain why it follows that $\displaystyle u^2 \notin \mathbb{Q} \Longrightarrow [\mathbb{Q}(u^2) \ : \ \mathbb{Q}] \ne 1 $

Peter