Outline

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