254A, Supplement 1: A little bit of algebraic number theory (optional)

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Analytic number theory is only one of many different approaches to number theory. Another important branch of the subject is algebraic number theory, which studies algebraic structures (e.g. groups, rings, and fields) of number-theoretic interest. With this perspective, the classical field of rationals $latex {{bf Q}}&fg=000000$, and the classical ring of integers $latex {{bf Z}}&fg=000000$, are placed inside the much larger field $latex {overline{{bf Q}}}&fg=000000$ of algebraic numbers, and the much larger ring $latex {{mathcal A}}&fg=000000$ of algebraic integers, respectively. Recall that an algebraic number is a root of a polynomial with integer coefficients, and an algebraic integer is a root of a monic polynomial with integer coefficients; thus for instance $latex {sqrt{2}}&fg=000000$ is an algebraic integer (a root of $latex {x^2-2}&fg=000000$), while $latex {sqrt{2}/2}&fg=000000$ is merely an algebraic number (a root of $latex {4x^2-2}&fg=000000$). For the purposes of this post, we will adopt the concrete (but…

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