Lectures for Mathematics 361, Spring 2025-26
- M 1/26: Overview: Z[i], sums of two squares, Pythagorean triples
- T 1/27: Overview: Quadratic character of 2, rational paametrization of
the circle, the t=tan(theta/2) substitution in calculus, cyclotomic fields
- W 1/28: Start Euclidean rings
- F 1/30: Finish Euclidean rings; start Euler's proof that \sum 1/p
diverges
- M 2/2: Arithmetic functions and Dirichlet series
- T 2/3: More arithmetic functions and Dirichlet series
- W 2/4: [Assignment A due]
Euclid's lemma, lcm, (Z/nZ,+,.), (Z/nZ)^x, Euler's rule, Fermat's
Little Theorem
- F 2/6: [Add/section-change/unit reduction
deadline]
ax+ny=b, ax=b(n); algorithms, Ruby code
- M 2/9: Sun-Ze Theorem
- T 2/10: Hensel's Lemma, start finitely generated abelian groups
- W 2/11: [Assignment B due]
Finish finitely generated abelian groups,
(Z/pZ)^\times, Z/p^eZ ~ Zp/p^eZp
- F 2/13: (Z/p^eZ)^\times for p odd, (Z/2^eZ)^\times, when is
(Z/nZ)^\times cyclic?
- M 2/16: The Bernoulli numbers, power sums, and zeta values
- T 2/17: The Bernoulli numbers, power sums, and zeta values
- W 2/18: [Assignment C due] Start
quadratic reciprocity--Euler's criterion, Gauss's lemma, (-5/p) by Gauss
- F 2/20: Legendre's formulation of QR, Euler QR <=> Legendre QR
- M 2/23: Jacobi symbol, start algebraic numbers and algebraic integers
- T 2/24: QR by Gauss sums: (2/p) and (p^*/q)=(q/p)
- W 2/25: [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 2/27: Zolotarev's proof of QR, start finite fields
- M 3/2: [No-W drop deadline] Finish
most of the finite fields writeup
- T 3/3: Finite fields loose ends, start characters
- W 3/4: [Assignment D due] Image and
kernel, N(x^e=u), orthogonality of characters, start Gauss sums
- F 3/6: Quick review of Gauss sums, Jacobi sum definitions and
diagonal counting formula, quadratic example, most of the table of
Jacobi sum values
- M 3/9: Finish Jacobi sum values, start cubic example and its modularity
- T 3/10: Finish cubic example and its modularity
- W 3/11: [Assignment E due]
Start arithmetic of D=Z[\omega]: unique factorization, units,
primes, factorization of rational primes, primary
primes
- F 3/13: (Informal meeting)
- M 3/16: Review arithmetic of D, cubic character
- T 3/17: Prove properties of the cubic character, state cubic
reciprocity, review proof of quadratic reciprocity
- W 3/18: Prove cubic reciprocity, examples in Z, sketch examples
beyond Z
- F 3/20: [Second quiz due]
Fermat for n=3, start Dirichlet's theorem on primes in arithmetic
progressions
- Spring break week
- M 3/30: Continue Dirichlet's theorem
- T 3/31: Continue Dirichlet's theorem
- W 4/1: [Assignment F due]
Continue Dirichlet's theorem
- F 4/3: (No meeting - Qual break)
- M 4/6: [Withdraw/leave deadline]
Finish Dirichlet's theorem on a.p.'s, preview rest of semester
- T 4/7: Start continuations and functional equations: Fourier
transform and the Gaussian, transformation law of theta
- W 4/8: [Assignment G due]
Finish continuation and functional equation of zeta
- F 4/10: Start continuation and functional equation for Dirichlet L-functions
- M 4/13: zeta as product of local integrals, review Dirichlet L and
continue onward through most of the argument
- T 4/14: Finish continuation and functional equation for Dirichlet
L-functions, skim Riemann-von Mangoldt
- W 4/15: [Assignment H due]
Quadratic fields, through norm
- F 4/17: More on quadratic fields, through quadratic character
- M 4/20: Review through quadratic character, start factorization of ideals
- T 4/21: Factorization of ideals
- W 4/22: More on quadratic fields, through the geometry section
- F 4/24: More on quadratic fields, through L(1,\chi)
- M 4/27: Quadratic fields: lattice points in a disk,
estimate A_n for Dedekind zeta
- T 4/28: Finish imaginary quadratic field class number formula
- W 4/29: (No meeting)
- F 5/1: (No meeting)
- W 5/13: [Term project due at 5pm]
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