Pries: 405 Number Theory, Spring 2012
Review sheet.
Topics to review
* = this topic will definitely be on the final.
1. primes, Euclidean algorithm, Bezout’s Lemma (ax + by = 1), unique factorization, Z
and Z/p[x].
2. units and multiplicative inverses, Euler’s φ-function, Chinese remainder theorem, zerodivisors.
√
3. Quadratic rings: Z[i], Z[ d], Pell’s equation.
4. * Modular arithmetic: Fermat’s Little Theorem, Euler’s Theorem, primitive roots.
5. * Quadratic reciprocity: definition of a square, Euler’s criterion, computing whether a
is a square modulo p, the theorem of quadratic reciprocity.
6. * Elliptic curves, group law, rational points, torsion points, elliptic curves over Z/p.
7. * Projective space, points at ∞.
8. * Finite fields: definitions and examples, finding inverses, cosets.
Topics that will not be on the exam
1. Sage.
2. Gauss sums.
3. Cryptography: Pollard’s p − 1 factoring, key exchange, RSA, El Gamal, discrete log,
elliptic curve cryptography.
4. Lattices, elliptic curves isomorphic to C/L.
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