ExampleConstruction: AI-adaptedVerification: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-12

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When char(k) is not 2, the affine circle x^2 + y^2 = 1 is birational to the affine line

Example

Assume $char (k) \neq 2$ , and let $C : = V (x^{2} + y^{2} - 1) \subseteq A_{k}^{2} .$ The line through $(- 1, 0)$ of slope $t$ meets $C$ again at $x = \frac{1 - t^{2}}{1 + t^{2}}, y = \frac{2 t}{1 + t^{2}} .$ This gives a rational map $A^{1} ⇢ C$ .

On the dense open subset of $C$ where $x \neq - 1$ , the inverse parameter is $t = \frac{y}{x + 1} .$ Substituting the displayed parametrization shows that these two formulas are mutually inverse on the common domain where $1 + t^{2} \neq 0$ and $x + 1 \neq 0$ . Thus the affine circle is birational to the affine line.

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17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, Algebraic Geometry, birational examples in Chapter 5l (standard reference, not scraped)
Donu Arapura, Notes on Basic Algebraic Geometry, rational-function examples in §3.1-§3.2 (standard reference, not scraped)