Euclidean proofs for function fields

Abstract

Schur proved the infinitude of primes in arithmetic progressions of the form $\equiv l\mod m$, such that $l^{2}\equiv1\mod m$, with non-analytic methods by ideas inspired from the famous proof Euclid gave for the infinitude of primes. Ram Murty showed that Schur's method has its limits given by the assumption Schur made. We will discuss analogous for the primes in the ring $\mathbb{F}_{q}[T]$.