Let $K$ be a number field. I was wondering how we know that there are infinite number of degree $1$ principal prime ideal of $K$.

Context: This is related to an example of polynomial representing infinitely many primes. Let $K$ be of degree $n$ and $\beta_1, ..., \beta_n$ be a $\mathbb{Z}$-basis of the ring of integers. Then the fact that $$ N_{K/\mathbb{Q}} ( \beta_1 X_1 + ... +\beta_n X_n ) \in \mathbb{Z}[X_1, ..., X_N] $$ represents infinite number of primes follows from the above fact. Thank you very much!

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    $\begingroup$ It follows from the existence of a pole at s=1 of the Dedekind zeta function. $\endgroup$ – franz lemmermeyer Dec 12 '15 at 16:38
  • $\begingroup$ @franzlemmermeyer Could you possibly elaborate on it by any chance? Thanks! $\endgroup$ – Johnny T. Dec 12 '15 at 21:05
  • $\begingroup$ This is usually proved in the classical approach to class field theory; see e.g. the recent book by Nancy Childress or, if you dare, Algebraic Numbers by Serge Lang. $\endgroup$ – franz lemmermeyer Dec 14 '15 at 17:12

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