 

Andrew J. Hetzel and Eric B. Morgan
A prime number theorem for finite Galois extensions view print


Published: 
August 10, 2015

Keywords: 
Binary quadratic form, Chebotarev's density theorem, field norm, Galois extension, prime number theorem 
Subject: 
Primary: 11R44; Secondary: 11D57, 11R11, 11R45 


Abstract
Let F be an algebraic number field and let P_{F}(r) denote the number of nonassociated prime elements of absolute field norm less than or equal to r in the corresponding ring of integers. Using information about the absolute field norms of prime elements and Chebotarev's density theorem, we readily show that when F is a Galois extension of Q, it is the case that P_{F} is asymptotic to (1/h)π, where π is the standard primecounting function and h is the class number of F. Along the way, we pick up some wellknown facts on the realizability of certain prime numbers in terms of those binary quadratic forms associated with the field norm over a ring of integers that is a unique factorization domain.


Author information
Andrew J. Hetzel:
Department of Mathematics, Tennessee Tech University, Cookeville, TN 38505, USA
ahetzel@tntech.edu
Eric B. Morgan:
Department of Mathematics, Tennessee Tech University, Cookeville, TN 38505, USA
emorgan@tntech.edu

