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Could someone recomend a book on number theory that uses a lot of group theory and algebra to explain results? Like for example the proof that mod that is a prime power has a primitive root is simple if we observe $\mathbb Z_{p^\alpha}^{\times}\cong \mathbb Z_{(p-1)p^{\alpha-1}}$. Also noticin half of the elements $\bmod p$ are quadratic residues is immediate if we look at it as a cyclic group. Or as a final example fermat and euler are immediate consequences of lagrange.

I'm looking for a book that treats results from this perspective.

Thank you very much in advance.

Regards.

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  • $\begingroup$ There are very few, probably because number theory is used to motivate abstract algebra so is usually taught first. They could also be taught in parallel, which may prove more illuminating. What's worse, many abstract algebra textbooks fail to explicitly point out how they generalize number theory results. $\endgroup$ – Bill Dubuque Feb 18 '15 at 2:13
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I highly suggest A Classical Introduction to Modern Number Theory, which uses algebra (and analysis upon occasion) whenever convenient.

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Elementary Number Theory: An Algebraic Approach by Ethan Bolker immediately comes to mind. It's got a long list of errata, though.

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