Lectures for Mathematics 361, Spring 202324
 M 1/22: Overview: Z[i], sums of two squares
 T 1/23: Pythagorean triples; start Euclidean rings
 W 1/24: More Euclidean rings
 F 1/26: Finish Euclidean rings; start Euler's proof that \sum 1/p
diverges
 M 1/29: Arithmetic functions and Dirichlet series
 T 1/30: More arithmetic functions and Dirichlet series
 W 1/31: [Assignment A due]
Euclid's lemma, lcm, (Z/nZ,+,.), (Z/nZ)^x, Euler's rule, Fermat's
Little Theorem
 F 2/2: [Add/sectionchange/unit reduction
deadline]
ax+ny=b, ax=b(n); algorithms, Ruby code
 M 2/5: SunZe Theorem
 T 2/6: Hensel's Lemma, start finitely generated abelian groups
 W 2/7: [Assignment B due]
Finish finitely generated abelian groups,
(Z/pZ)^\times, Z/p^eZ ~ Zp/p^eZp
 F 2/9: (Z/p^eZ)^\times for p odd, (Z/2^eZ)^\times, when is
(Z/nZ)^\times cyclic?
 M 2/12: The Bernoulli numbers, power sums, and zeta values
 T 2/13: The Bernoulli numbers, power sums, and zeta values
 W 2/14: [Assignment C due] Start
quadratic reciprocityEuler's criterion, Gauss's lemma, (5/p) by Gauss
 F 2/16: Legendre's formulation of QR, Euler QR <=> Legendre QR
 M 2/19: Jacobi symbol, start algebraic numbers and algebraic integers
 T 2/20: QR by Gauss sums: (2/p) and (p^*/q)=(q/p)
 W 2/21: [First quiz due]
Review QR by Gauss sums, gloss the polyomial of the sum or product
of algebraic numbers by resultants, gloss sign of the Gauss sum by
Fourier analysis
 F 3/23: Zolotarev's proof of QR, start finite fields
 M 2/26: [NoW drop deadline] Finish
most of the finite fields writeup
 T 2/27: Finite fields loose ends, start characters
 W 2/28: [Assignment D due] Image and
kernel, N(x^e=u), orthogonality of characters, start Gauss sums
 F 3/1: Quick review of Gauss sums, Jacobi sum definitions and
diagonal counting formula, quadratic example, most of the table of
Jacobi sum values
 M 3/4: Finish Jacobi sum values, start cubic example and its modularity
 T 3/5: Finish cubic example and its modularity
 W 3/6: [Assignment E due]
Start arithmetic of D=Z[\omega]: unique factorization, units,
primes, factorization of rational primes, primary
primes
 F 3/8: (Informal meeting)
 Spring break week
 M 3/18: Review arithmetic of D, cubic character
 T 3/19: Prove properties of the cubic character, state cubic
reciprocity, review proof of quadratic reciprocity
 W 3/20: Prove cubic reciprocity, examples in Z, sketch examples
beyond Z
 F 3/22: [Second quiz due]
Fermat for n=3, start Dirichlet's theorem on primes in arithmetic
progressions
 M 3/25: Continue Dirichlet's theorem
 T 3/26: Continue Dirichlet's theorem
 W 3/27: [Assignment F due]
Continue Dirichlet's theorem
 F 3/29: (No meeting  Qual break)
 M 4/1: [Withdraw/leave deadline]
Finish Dirichlet's theorem on a.p.'s, preview rest of semester
 T 4/2: Start continuations and functional equations: Fourier
transform and the Gaussian, transformation law of theta
 W 4/3: [Assignment G due]
Finish continuation and functional equation of zeta
 F 4/5: Start continuation and functional equation for Dirichlet Lfunctions
 M 4/8: zeta as product of local integrals, review Dirichlet L and
continue onward through most of the argument
 T 4/9: Finish continuation and functional equation for Dirichlet
Lfunctions, skim Riemannvon Mangoldt
 W 4/10: [Assignment H due]
Quadratic fields, through norm
 F 4/12: More on quadratic fields, through quadratic character
 M 4/15: More on quadratic fields, through fundamental domain
 T 4/16: More on quadratic fields, through lattice results
 W 4/17: More on quadratic fields, through L(1,\chi)
 F 4/19: Quadratic fields: lattice points in a disk,
estimate A_n for Dedekind zeta
 M 4/22: Finish ideal class number formula for imaginary
quadratic fields; mention cyclotomic zeta
 T 4/23: Start real quadratic units
 W 4/24: More real quadratic units, evaluations
 F 4/26: (No meeting)
 M 5/6: [Term project due at noon]
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