Infinitely many primes congruent to 1 modulo n

Statement

Let $n$ be a positive integer. Then there are infinitely many prime numbers $p$ such that $p \equiv 1 \pmod{n}$.

Remarks

This is the case $a = 1$ of Dirichlet's theorem on arithmetic progressions. The case $a = 1$ has an elementary proof with cyclotomic polynomials. A prime factor $p$ of $\Phi_n(m)$ that does not divide $n$ satisfies $p \equiv 1 \pmod n$. For example, the primes $p \equiv 1 \pmod{10}$ start with $11, 31, 41, 61, 71$.

Explanations

L1 human

Fix a number $n$, for example $n = 6$. Then infinitely many primes leave the remainder $1$ when you divide them by $6$. The first ones are $7, 13, 19, 31, 37$.