A basis in $\mathbb{Q}(\pi)$

Asked Jun 05 '23 at 20:12

Active Jun 05 '23 at 23:16

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I want to find a basis in $\mathbb{Q}(\pi)$.

What I've tried: $\mathbb{Q} \subset \mathbb{R}$ is a field extension, and $\pi \in \mathbb{R}$ is transcendental in $\mathbb{Q}$. Then $\mathbb{Q}(\pi) \simeq \mathbb{Q}(X)$ and, because $1, X, X^2, \dots, X^n, \dots$ is a basis in $\mathbb{Q}$, then $1, \pi, \pi^2, \pi^3, \dots, \pi^n, \dots$ is a basis in $\mathbb{Q}(\pi)$. Is this true? Thanks!

edited Jun 05 '23 at 20:14

Chickenmancer

4,347

asked Jun 05 '23 at 20:12

MathLearner

5

How do you represent $\frac1{\pi}$ with your basis? – Christian E. Ramirez Jun 05 '23 at 20:16
2

But you wrote $\Bbb Q(X)$, not $\Bbb Q[X]$. – Ted Shifrin Jun 05 '23 at 20:16
@TedShifrin oh. Yes. I understand now. But what's a basis in $\mathbb{Q}(X)$ ? – MathLearner Jun 05 '23 at 20:23

A basis in $\mathbb{Q}(\pi)$

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