infinite primes $p\equiv1\pmod n$ without cyclotomic polynomial

Question

Without cyclotomic polynomial, is there an elementary proof of the following: for each integer $n>1$, there are infinitely many primes $p$ such that $p\equiv1\pmod n$ ?

please don't refer to Dirichlet's theorem on arithmetic progressions, or analytic number theory. Thanks

Answer 1

An elementary proof of this result is outlined in problem 36 on p. 108 of An Introduction to the Theory of Numbers by Niven, Zuckerman and Montgomery. That problem references the article by I. Niven and B. Powell "Primes in certain arithmetic progressions," Amer. Math. Monthly, 83 (1976), 467-469, as simplified by R.W. Johnson.