*Meeting 14: December 3*

Back to quadratic reciprocity, quadratic forms, and cyclotomy

- Recap: algebraic integers, quadratic and cyclotomic extensions
- Recap: unique factorization of ideals
- Primes splitting and ramifying, and Fermat's theorem
- The class group
- The correspondence between quadratic forms and ideals
- Gauss 6

Characterizing structuralism

- Language
- Effects on definitions and concepts (modding out by irrelevant detail, giving invariants algebraic relevance)
- Effects on proofs (controlling indifference types, managing context)
- Benefits: reducing cognitive burdens, modularity, maintaining focus, determining relevance, outlinability, generalizability

Contrast case studies

- Historical case studies: want clear and striking contrasts
- Need to identify clear limitations and failures vs. successes
- An example: quadratic forms and cyclotomy from Gauss to Dedekind / Hilbert; Kronecker-Weber explains periodicity

*Meeting 13: November 19*

Dedekind's theory of ideals

- Recap: algebraic integers
- The behavior of 2 in Z[sqrt -5]
- The need for gcd's
- Ideals in a ring of integers
- The unique factorization theorem
- Calculations in Z[sqrt -5]
- Proof of unique factorization in quadratic extensions

*Meeting 12: November 12*

Gauss 6 and cyclotomic extensions

- Cyclotomic fields, mod q
- Gaussian periods
- Gauss sums and the sixth proof, revisited

The "meaning" of a number-theoretic fact

- Meaning via definitions
- Meaning as use; inferential aspects
- Arana-Detlefsen purity concerns
- Concerns about stability of meaning over time

*Meeting 11: November 5*

Galois theory: comparison of Galois' and the modern approach

- Overview of Galois' approach
- Small differences: notions get polished, account is more general
- Big difference: eliminating talk of resolvent
- Overview of the modern approach
- Proof that the fixed field of the Galois group of a normal extension is the original field
- Discussion: definitions in terms of algebraic structures; "automorphisms" independent of representation

Theory of algebraic integers

- Algebraic number fields in number theory: quadratic forms, cyclotomic fields (quadratic reciprocity, Fermat's last theorem)
- Quote by H. J. S. Smith
- Historical overview
- Motivating the definition of "algebraic integer"
- Inferring "ideal" factors in Z[sqrt -5]

*Meeting 10: October 29*

Finite fields

- Factorization of polynomials in finite fields
- The Frobenius automorphism and Galois group of a finite field

Algebraic number fields, modulo q

- Z[x]/(p(x)), p irreducible; modulo q
- p may factor mod q; Z[x]/(p_i(x),q) isomorphic to F_{q^n}
- Pulling the Frobenius back to the Galois group of Q[x]/(p(x))

Specializing to cyclotomic fields, p(x) = (x^p-1)/(x - 1)

- alpha -> alpha^q permutes roots
- order depends only on order of q in (Z/pZ)^*
- Gauss' determination of S^2 means that Q(sqrt p) is a subfield
- Frobenius maps sqrt p to +- sqrt p

*Meeting 9: October 22*

Properties of the Galois group

- recap: properties of field extensions, definition of Galois group
- each element really is a permutation of the roots
- polynomial expressions in the roots that are fixed by elements of the Galois group
- elements give rise to automorphisms
- Galois group doesn't depend on choice of resolvent
- Galois group really is a group
- The field extension is normal

Moving to higher reciprocity

- Higher reciprocity laws - break symmetry between p and q
- Finite fields and extensions to splitting fields

*Meeting 8: October 15*

Galois' presentation of the Galois group

- recap: the cubic formula, and primitive 11th roots of unity
- overview of Galois' approach
- interlude: discussion of quote on p. 240 of Tignol. Issues:
- managing / controlling calculation
- moving from particular to general (e.g.~from a particular cyclotomic extension to arbitrary ones)
- avoiding (non-canonical) representations
- getting to the "essence" (Dedekind: "fundamental characteristics")

- background from modern algebra: constructing algebraic extensions
- existence of the Galois resolvent
- the Galois group

Discussion of Gauss' sixth proof

- calculating in a field extension vs. calculating modulo a polynomial
- operating, as it were, with a single root, but behind the veil of symmetry
- interpreting Gauss' lemmas

*Meeting 7: October 8*

Discussion of Sandborg's thesis

- overview and taxonomy of proofs (inductive, Euler-based, etc.)
- inductive proofs: additivity and base case
- distinction between algebra and geometry
- Niven and Zuckerman proof and SL(2,z)
- From explanation to intelligibility virtues

Gauss sums

- Gauss 4 and 6 generalize to higher reciprocity laws
- Overview of Gauss 4
- Eisenstein's simplification (his second proof)

*Meeting 6: October 1*

Views on calculation: Euler, Gauss, Galois, Dedekind, Kronecker, Siegal, Abhyankar, Edwards

Composition of binary quadratic forms

Galois theory

- Prehistory: quadratic, cubic, quartic, Lagrange, Vandermonde, Gauss, Abel.
- Solution to the cubic (and Lagrange's analysis)
- Vandermonde's method of obtaining a primitive 11th root of unity

*Meeting 5: September 24*

More on quadratic forms

- Representability and quadratic residues
- A reduction theorem for positive definite forms
- An algorithm for determining representability
- Composition of forms

Quadratic reciprocity

- On the list of 233 proofs, about 63 are based on Gauss' lemma; many based directly on Gauss #3.
- The calculations in Gauss #3
- Eisenstein's proof
- Comparison: focus on parity, focus on up-to-parity schematic indications
- Coming up: Gauss sums

*Meeting 4: September 17*

Back to aspects of mathematical understanding

- Recall: substance, control, transparency
- Conventional logic: focus on ultimate result and everything involved in getting it
- One aspect of transparency: stage assessments, using means of expression to cordon off detail
- Example: Hartley Rogers' axioms for computability
- Another aspect: uniformity of treatment
- Suggestive: mode of expression "brings out" a uniformity already known to experts, makes it shareable
- Distinction between two senses of "generality": uniformity of treatment, scope

Galois on computation and elegance: html

Back to quadratic forms

- Composing substitutions
- Invariance of determinant
- Representability and quadratic residues

*Meeting 3: September 10*

Quadratic reciprocity

- The data: squares and nonsquares modulo p
- Euler's criterion
- Gauss #1: fiddly induction
- Gauss #3: based on Gauss' lemma (and a nifty graphical representation)

Quadratic forms

- Overview
- Proof of Fermat's theorem using the Gaussian integers
- Equivalence of forms

*Meeting 2: September 3*

Sums of squares

- Background: Fermat's theorem, Lagrange's theorem
- Fermat's theorem on primes representable as sums of squares
- Reciprocity step
- Descent step (Euler's proof)

Aspects of mathematical understanding

- Concerns: indvidualism, subjectivity, historicity; need systematic, shareable criteria
- Substance (nontriviality, of proofs, of consequences)
- Cogency (rigor, precision, maintaining agreement)
- Transparency (surveyability, navigability, isolating key ideas)

*Meeting 1: August 27*

Philosophical issues

- Euler's 1747 letter on patterns in the prime: "understanding" the raw data
- The relatively recent focus on "certainty" in mathematics, and Cristopher Clavius
- Issues of stability in mathematics
- Frege and formal notions of proof
- 233 proofs of quadratic reciprocity
- Shifting focus from proof to forms of understanding

Historical overview

- The evolution from Gauss' DA to the Dirichlet-Dedekind supplements
- The world in 1801
- Mathematics prior to 1801
- Overview of the DA