Number Theory for BeginnersIn the summer quarter of 1949, I taught a ten-weeks introductory course on number theory at the University of Chicago; it was announced in the catalogue as "Alge bra 251". What made it possible, in the form which I had planned for it, was the fact that Max Rosenlicht, now of the University of California at Berkeley, was then my assistant. According to his recollection, "this was the first and last time, in the his tory of the Chicago department of mathematics, that an assistant worked for his salary". The course consisted of two lectures a week, supplemented by a weekly "laboratory period" where students were given exercises which they were. asked to solve under Max's supervision and (when necessary) with his help. This idea was borrowed from the "Praktikum" of German universi ties. Being alien to the local tradition, it did not work out as well as I had hoped, and student attendance at the problem sessions so on became desultory. v vi Weekly notes were written up by Max Rosenlicht and issued week by week to the students. Rather than a literal reproduction of the course, they should be regarded as its skeleton; they were supplemented by references to stan dard text-books on algebra. Max also contributed by far the larger part of the exercises. None of ,this was meant for publication. |
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a=b mod a=x+iy additive group André Weil apply theorem bijection binary operation called closed under addition common divisor commutative groups complex numbers congruence classes modulo congruence classes prime congruence relation congruence x²=a mod congruence x³=a corollary of theorem coset cyclic group defined Definition denoted element of G exercise V.6 factors finite G of order Gauss Gaussian prime Gaussian set group G group of congruence group of order group under multiplication Hint implies integer m>0 integer prime isomorphism Legendre symbol lemma Let G linear combinations logn mutually relatively prime neutral element norm number of pairs Number Theory ordinary integers pairs x,y polynomial of degree positive integers primitive root modulo product of primes proof prove by induction quadratic reciprocity law quadratic residue modulo rational numbers rational prime real numbers root of unity square-free subgroup of G theorem II.1 unique solution uniquely determined write x=y mod zero-divisors