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Dirichlet arithmetic progression theorem, or more generally, Chabotarev density theorem, has applications to algebraic number theory, especially in class-field theory.
Since we might think of the density theorem as an analytic theorem, and as prime number theorem is one main theorem of analytic number theory, one is led to wonder:

if there is any application of prime number theorem to algebraic number theory.

Thanks for any attention in advance.

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There is a prime ideal theorem for number fields, but I guess that's more of a generalization than an application. –  Gerry Myerson Feb 13 '13 at 12:22
Do you refer to this?Thanks for pointing this out. –  awllower Feb 13 '13 at 12:59
Yes, that's what I had in mind. –  Gerry Myerson Feb 13 '13 at 23:08
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I don't know if this can be considered algebraic number theory or if it is more algebraic geometry; but, here goes.

Deligne uses the methods of Hadamard-de la Vallée-Poussin to prove the Weil conjectures. Even if it is not an application of the ordinary PNT as such, the exact same methods are applied elsewhere.

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I see your point: PNT leads to breakthroughs in other fields. Even though this is not what I expected(I was expecting something analytic.), I think this is a good answer. :) –  awllower Oct 20 '13 at 4:18
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