Rational-function field F_5(t)

The rational-function field $\mathbb F_5(t)$ is the fraction field of the polynomial ring $\mathbb F_5[t]$ . An element is a quotient $a(t)/b(t)$ , with $a,b\in\mathbb F_5[t]$ and $b\ne0$ ; two fractions agree when cross multiplication gives the same polynomial.

Global and local roles

This is a global function field. Its places include those attached to monic irreducible polynomials and the place at infinity. Completing at $(t)$ gives $\mathbb F_5((t))$ , which is a different field.

Scalar conventions

The constants embed as $\mathbb F_5$ . A homomorphism over the constant field fixes these constants, while an abstract field map need not fix each of them when $q$ is not prime. No topology is part of this bare global-field record.