Transcendence of $\pi$ (awaiting a scope decision)

Transcendence of $\pi$. The number $\pi$ is transcendental over $\mathbb{Q}$: it is not a root of any nonzero polynomial with rational coefficients. In particular $\pi$ is irrational, and $\pi$ is not constructible by straightedge and compass, so the circle cannot be squared.

Status: settled, and awaiting an owner decision here. Lindemann proved it in 1882. Like The Lindemann-Weierstrass theorem (awaiting a scope decision) ‡, of which it is the headline corollary, it is neither open nor blocked by a missing track; it has been flagged for a decision on whether to author the transcendence machinery, and until that decision is taken it is recorded and not used.

Not proved in this library. No page here proves that $\pi$ is transcendental, and no proof in this library may lean on it.

What is known, and what would settle its place here. The derivation from The Lindemann-Weierstrass theorem (awaiting a scope decision) ‡ is short: if $\pi$ were algebraic then so would $i\pi$ be, and $e^{i\pi} = -1$ together with $e^0 = 1$ would exhibit two exponentials of distinct algebraic numbers that are linearly dependent over $\overline{\mathbb{Q}}$, contradicting the theorem. So the whole cost sits in the theorem, plus Euler's identity, which this library does intend to prove once the complex exponential is built. Authoring both is what would replace this item by a theorem.

Why it matters here. Transcendence and irrationality are routinely confused, and the difference is exactly the difference between what this library proves and what it records. The irrationality of $\sqrt{2}$ is proved here, and the irrationality of $\pi$ alone has a famously short elementary proof (Niven, 1947) that is well inside scope. Transcendence is a strictly stronger and strictly more expensive statement, and the classical geometric consequence, the impossibility of squaring the circle, needs the stronger one. Recording the distinction is the point of this item.