Baker — Algebra & Number Theory
Andrew Baker's third-year course notes at the University of Glasgow,
73 pages in four chapters, copyright 2009. The title page carries a second date, "09/01/2018," which
reads as a revision stamp and is ambiguous between 9 January and 1 September. No published: is
recorded for that reason: the copyright year gives a floor, not a publication date.
Tier T1: a working mathematician's own course notes, for a course he taught. Pitched at the third year of a UK degree, so a reader who has met proofs and induction but not rings or fields. Rigour is theorem-and-proof throughout, with numbered examples and four problem sets (45 numbered questions, no solutions in the file).
Contents
| Ch. | Subject | Pages |
|---|---|---|
| 1 | Basic number theory: induction and well-ordering, the Euclidean algorithm, congruences, primes, Fermat and Wilson, continued fractions, Diophantine equations, Pell's equation | 1–30 |
| 2 | Groups and group actions: permutations and their sign, symmetry groups, Lagrange, orbit-stabilizer, Burnside | 31–50 |
| 3 | Arithmetic functions: convolution and Möbius inversion | 51–56 |
| 4 | Finite and infinite sets: cardinality, countability, power sets, uncountability of ℝ | 57–66 |
See number-theory for the arithmetic chapters, group-theory for chapter 2 and cardinality-and-countability for chapter 4.
What is unusual about it
Number theory comes first and needs no rings. Both algebra texts already held here (lal-algebra-1, judson-abstract-algebra) reach arithmetic after introducing rings, on the argument that the ring is the right language for it. Baker does it the other way round: the integers, the Euclidean algorithm and congruences are done by hand in chapter 1, and the group theory in chapter 2 then pays back the debt. Theorem 1.27 (a primitive root exists modulo a prime) is stated early and proved in chapter 2 as a case of a general fact, that a finite group with at most d solutions of x^d = ι for every divisor d of its order is cyclic. The proof counts elements of each order against the Euler function φ.
Applications are internal. Chapter 2's payoff is Burnside's formula, used to count distinguishable colourings of symmetric objects. The notes never mention cryptography, RSA or any computational use of the number theory. Fermat's little theorem, the Chinese remainder theorem and primitive roots are all here, which is the toolkit judson-abstract-algebra chapter 7 applies to public-key cryptography, but that application is not made.
The last chapter is a different subject. Cardinality and countability end the course with Cantor's diagonal argument. It is set theory, and the notes give no reason for its place beyond the opening remark that the natural numbers arose from counting sets.
Cross-spoke context
Runner-up ../cryptography-wiki: considered and not ingested there. The notes teach the number
theory that cryptography uses and make no cryptographic claim, which is the boundary rule in
../CLAUDE.md: the mathematics is the subject here, the application is the subject there. Recorded as
context only.
Reading caveat
Text extraction garbles displayed equations and drops some symbols. The summary rests on section headings, theorem statements and proofs read in prose, not on any formula.