Euclidean proofs for function fields. (English) Zbl 1410.11126

Lachmann, Thomas

Funct. Approximatio, Comment. Math. 58, No. 1, 105-116 (2018).

Summary: Schur proved the infinitude of primes in arithmetic progressions of the form \(\equiv l\operatorname{mod} m\), such that \(l^{2}\equiv1\operatorname{mod} m\), 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 \(\mathbb{F}_{q}[T]\).

MSC:

11R58

Arithmetic theory of algebraic function fields

Keywords:

Euclidean proof; function fields; Carlitz module

References:

[1]	S. Bae, S.-G. Hahn, On the ring of integers of cyclotomic function fields, Bull. Korean Math. Soc. 29 (1992), 153-163. · Zbl 0765.11046
[2]	A.S. Bamunoba, Arithmetic of Carlitz polynomials, https://scholar.sun.ac.za/handle/10019.1/95850 [17.02.2016]. · Zbl 1385.11038
[3]	M.F. Becker, S. Maclane, The minimum number of generators for inseparable algebraic extensions, http://projecteuclid.org/download/pdf_1/euclid.bams/1183502442 [17.02.2016]. · Zbl 0022.30402
[4]	K. Conrad, Euclidean proofs of Dirichlet’s theorem, http://www.math.uconn.edu/ kconrad/blurbs/gradnumthy/dirichleteuclid.pdf [17.02.2016].
[5]	B. Hornfeck, Primteiler von Polynomen, J. Reine Angew. Math. 243 (1970), 120.
[6]	S. Jeong, Resultants of cyclotomic polynomials over \(\mathbb{F}_{q}[T]\) and applications, Commun. Korean Math. Soc. 28 (2013), 25-38. · Zbl 1306.11085
[7]	M.R. Murty, N. Thain, Prime numbers in certain arithmetic progressions, Functiones et Approximatio XXXV (2006), 249-259. · Zbl 1196.11011
[8]	M. Rosen, Number Theory in Function Fields, Springer-Verlag, New York 2002. · Zbl 1043.11079
[9]	H. Stichtenoth, Algebraic Function Fields and Codes, Springer-Verlag, Berlin Heidelberg 2009. · Zbl 1155.14022