October 21
Lecture 26
There are infinitely many primes: Euclid's idea
continued fractionsprimes
What we covered
Lectures
This page records what we covered after each class meeting, with links to the corresponding lecture notes.
October 19
Lecture 25
Solving Pell's equations using continued fractions
Pell's equationscontinued fractions
October 16
Lecture 24
Units in \(\mathbb{Z}[\sqrt{2}]\)
unitsPell's equations
October 14
Lecture 23
Primes and units in \(\mathbb{Z}[\sqrt{\pm 2}]\)
primesunitsPell's equations
October 12
Lecture 22
Pythagorean triples
Pythagorean triples
October 9
Midterm 1
Midterm 1
October 7
Lecture 21
Primes in \(\mathbb{Z}[i]\) (cont.)
primesGaussian integers
October 5
Lecture 20
Review 1
Unique factorization in\(\mathbb{Z}\)quadratic residues and quadratic reciprocityprime factorization in\(\mathbb{Z}[\sqrt{d}]\)
October 2
Lecture 19
Units and primes in \(\mathbb{Z}[i]\) (cont.)
unitsprimesGaussian integers
September 30
Lecture 18 (Zoom)
Units and primes in \(\mathbb{Z}[i]\)
unitsprimesGaussian integers
September 28
Cancelled
Units and primes in \(\mathbb{Z}[i]\)
unitsprimesGaussian integers
September 25
Lecture 17
Unique factorization in \(\mathbb{Z}[\sqrt{d}]\) for \(d=-1,\pm2,3\). III.
Bézout's lemmaEuclid's lemmaunique factorization
September 23
Lecture 16
Unique factorization in \(\mathbb{Z}[\sqrt{d}]\) for \(d=-1,\pm2,3\). II.
division algorithmPIDgcdBézout's lemma
September 21
Lecture 15
Unique factorization in \(\mathbb{Z}[\sqrt{d}]\) for \(d=-1,\pm2,3\). I.
division algorithm
September 18
Lecture 14
Existence of prime factorization and examples of nonunique factorization in \(\mathbb{Z}[\sqrt{d}]\)
prime factorization in\(\mathbb{Z}[\sqrt{d}]\)
September 16
Lecture 13
Arithmetic in \(\mathbb{Z}[\sqrt{d}]\) (cont.)
normunitsprimes
September 14
Lecture 12
Proof of the sum of two squares theorem (complete), arithmetic in \(\mathbb{Z}[\sqrt{d}]\)
The ring\(\mathbb{Z}[\sqrt{d}]\)Gaussian integersdivisibilityunitsprimes and compositesnormunique factorization
September 11
Lecture 11
Sum of two squares
sum of two squares
September 9
Lecture 10
Proof of quadratic reciprocity
quadratic reciprocityJacobi symbolsum of two squares
September 4
Lecture 9
The law of quadratic reciprocity
quadratic reciprocitycounting lattice points
September 2
Lecture 8
Gauss's lemma (cont.)
Gauss's lemma\(2 \pmod{p}\)
August 31
Lecture 7
Gauss's lemma
Wilson's theoremfloor functionGauss's lemma
August 28
Lecture 6
Euler's criterion and its applications
quadratic residues & nonresiduesEuler's criterionLegrendre symbol\(-1 \pmod{p}\)
August 26
Lecture 5
Quadratic residues and nonresidues
polynomial congruencesLagrange's theoremquadratic congruencesquadratic residues & nonresidues
August 24
Lecture 4
Euler's theorem and linear congruence equations
residue classesEuler's totient functionEuler's theoremFermat's little theoremlinear congruences
August 21
Lecture 3
Review of unique factorization (complete), review of Euclidean algorithm and congruences
FTAUFDEuclidean domainEuclidean algorithmcongruences
August 19
Lecture 2
Review of unique factorization (cont.)
FTAidealsPIDgcdBézout's lemmaEuclid's lemma
August 17
Lecture 1
Course introduction and review of unique factorization
syllabusdivisibilityunitsprimes and compositesFTAdivision algorithm
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