### Lectures for Mathematics 361, Spring 2023-24

• 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

• M 2/5: Sun-Ze 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 reciprocity--Euler'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: [No-W 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 L-functions

• 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 L-functions, skim Riemann-von Mangoldt
• W 4/10: [Assignment H due] Quadratic fields, through norm

• 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]