Abstract Algebra¶
Job: structure theory of groups, rings, modules, and field extensions as used in number theory, geometry, and representation theory.
Core¶
Algebra organizes mathematical objects by operations and the maps that preserve them. The standard pipeline for quals and research preliminaries:
| Layer | Objects / maps | Key theorems |
|---|---|---|
| Groups | \(G\), \(H\le G\), \(N\trianglelefteq G\), \(\varphi:G\to G'\) | Lagrange, Sylow, structure of f.g. abelian groups, Jordan–Hölder |
| Rings / fields | \(R\), ideals, \(R/\mathfrak{a}\), \(R_{\mathfrak{p}}\), \(\operatorname{Spec} R\) | Noetherian, localization exactness, PID \(\Rightarrow\) UFD |
| Modules | \(M\in R\text{-Mod}\), \(\operatorname{Hom}\), \(\otimes\) | structure over PID; \(\operatorname{Hom}\) left exact, \(\otimes\) right exact |
| Galois | \(L/K\) finite Galois, \(\operatorname{Gal}(L/K)\) | FTGT; solvability \(\Leftrightarrow\) radical extensions (char \(0\)) |
Dependency order: groups \(\to\) rings \(\to\) modules \(\to\) Galois (field theory uses both rings and groups).
Reading map¶
| Page | Quals focus |
|---|---|
| Group theory | actions, Sylow, f.g. abelian, exact sequences |
| Rings and fields | Noetherian, localization, \(\operatorname{Spec}\) intuition |
| Modules | free/projective/injective; Hom/Tensor exactness |
| Galois theory | FTGT, cyclotomic, solvability by radicals |
| Commutative algebra | Spec, dimension, depth, Serre criteria |
| Representation theory | Characters, Lie theory |
| Homological algebra | Ext/Tor, derived categories |
| Algebraic geometry | Spec, schemes, coherent cohomology |
| Category theory | Universality of \(\operatorname{Hom}\), \(\otimes\), free functors |
Signature theorems (orientation)¶
- Lagrange / Sylow: \(|H|\mid|G|\); Sylow \(p\)-subgroups exist and are conjugate; \(n_p\equiv1\pmod p\), \(n_p\mid|G|/p^k\).
- Finitely generated abelian groups: \(G\cong\mathbb{Z}^r\oplus\bigoplus\mathbb{Z}/n_i\) (invariant factors or elementary divisors).
- Isomorphism theorems: \(G/\ker\phi\cong\operatorname{im}\phi\); correspondence and diamond theorems.
- Hilbert basis: \(R\) Noetherian \(\Rightarrow R[x]\) Noetherian.
- FTGT: for finite Galois \(L/K\), subgroups \(H\le\mathrm{Gal}(L/K)\) \(\leftrightarrow\) intermediate fields \(L^H\); normal subgroups \(\leftrightarrow\) Galois subextensions.
- Structure over PID: f.g. modules \(\cong R^r\oplus\bigoplus R/(d_i)\) — Jordan/rational form of linear maps is the \(K[t]\) case.
Notation (paper-standard)¶
- \(\operatorname{Hom}(G,H)\), \(\operatorname{Aut}(G)\), \(\operatorname{Inn}(G)\), \(G^{\mathrm{ab}}=G/[G,G]\)
- \(\mathfrak{p},\mathfrak{m}\) prime/maximal ideals; \(R^\times\) units; \(\operatorname{Spec} R\)
- \(M\otimes_R N\), \(\operatorname{Tor}_i^R\), \(\operatorname{Ext}_R^i\)
- \(\operatorname{Gal}(L/K)\), fixed field \(L^H\); \([L:K]=|\mathrm{Gal}|\) when Galois
Interfaces¶
| Area | Contact |
|---|---|
| Number theory | \((\mathbb{Z}/n\mathbb{Z})^\times\); cyclotomic Galois |
| Algebraic NT | \(\mathcal{O}_K\), Dedekind, class group as module theory |
| Homological algebra | Derived functors on \(R\)-Mod |
| Representation theory | \(KG\)-modules; characters |
See it¶
Class field theory and étale cohomology sit on top of: Galois groups of infinite extensions (\(\operatorname{Gal}(\overline{K}/K)\) as inverse limit), Dedekind domains / \(\operatorname{Spec}\mathcal{O}_K\), and cohomology of modules (\(H^1(G,M)\)). Even without that package, the same lemmas (Sylow, Nakayama, FTGT) appear in every representation-theory or arithmetic paper’s preliminaries.
Nakayama’s lemma (use-case). If \(M\) is f.g. over local \((R,\mathfrak{m})\) and \(M=\mathfrak{m}M\), then \(M=0\). Generates: minimal generating sets lift from \(M/\mathfrak{m}M\); free modules detected by rank of fibers — ubiquitous in deformation theory and local algebra.
Check¶
Why is \(\mathbb{Z}/4\mathbb{Z}\not\cong\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}\) as groups, yet both are rings of order \(4\)?
As groups: unique element of order \(4\) vs all non-identity elements order \(\le 2\). As rings they are also non-isomorphic (\(x^2=0\) nilpotents, units, etc.). Structure theorems need the full algebraic type.
If \(0\to A\to B\to C\to 0\) is exact in Ab, is \(0\to\operatorname{Hom}(C,M)\to\operatorname{Hom}(B,M)\to\operatorname{Hom}(A,M)\to 0\) exact?
Left exact yes; surjectivity on the right fails in general (extension classes live in \(\operatorname{Ext}^1(C,M)\)). Dual: \(\otimes\) is right exact, not left.
State the lattice form of the Galois correspondence for finite Galois \(L/K\).
Inclusion-reversing bijection \(\{H\le G\}\leftrightarrow\{K\subseteq F\subseteq L\}\) via \(H\mapsto L^H\), \(F\mapsto\mathrm{Gal}(L/F)\), with \(|H|=[L:L^H]\) and \([F:K]=[G:\mathrm{Gal}(L/F)]\).
Pitfalls¶
- “Normal \(\Leftrightarrow\) quotient is a group” is correct; “all subgroups normal” only for Dedekind groups / abelian — not \(S_n\).
- PID \(\neq\) Euclidean: \(\mathbb{Z}[\frac{1+\sqrt{-19}}{2}]\) is a classic PID that is not Euclidean.
- Galois correspondence requires normality of the big extension (or work with intermediate normal extensions carefully).
- Noetherian is not preserved by infinite products of rings; ascending chain condition is about ideals, not elements.
- Projective \(\neq\) free in general (\(R=\mathbb{Z}/6\mathbb{Z}\) examples; Dedekind domains: projective \(\Leftrightarrow\) locally free).
Refs¶
- Lang, Algebra
- Dummit–Foote, Abstract Algebra (computational / examples)
- Atiyah–Macdonald, Introduction to Commutative Algebra (rings/modules)
- Hungerford / Rotman (alternate comprehensive texts)
Next¶
Group theory · Rings and fields · Modules · Galois theory · Commutative algebra · Category theory · Number theory