Learning Path — Algebraic Number Theory & Class Field Theory
Suggested preparation for Algebraic Number Theory & Class Field Theory. Choose an entry point that fits your preparation.
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Entry / bridge route · Additional preparation for research reading · All paths
Entry / bridge route
Proofs & Mathematical Writing
Read quantified statements; write direct, contrapositive, contradiction and induction arguments.
Book of Proof — Richard Hammack. Direct open PDF
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You can explain the quantifiers in a statement and write these forms of proof without relying on a worked solution.
Elementary Number Theory
Use divisibility, congruences, Euclid’s algorithm and elementary reciprocity examples.
Elementary Number Theory: Primes, Congruences, and Secrets — William Stein. Direct open PDF
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You can prove elementary arithmetic statements and solve congruences, not just apply memorized formulas.
Linear Algebra
Work with vector spaces, bases, linear maps, kernels, rank and eigenvalues.
Linear Algebra — Jim Hefferon. Open online text / archive
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You can prove rank–nullity, change bases and distinguish diagonalizable maps from general linear maps.
Abstract Algebra & Galois Theory
Use groups, rings, ideals, quotient constructions and field extensions.
Abstract Algebra: Theory and Applications — Thomas W. Judson. Open online text / archive
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You can construct quotient groups and rings correctly and state the assumptions of the isomorphism theorems.
Commutative Algebra
Study modules, localization, prime ideals and Noetherian arguments.
A Primer of Commutative Algebra — J. S. Milne. Direct open PDF
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You can localize a ring or module, work with prime ideals and check exactness in basic module examples.
Algebraic Number Theory & Class Field Theory
Study number fields, rings of integers, ideals and local arithmetic.
Algebraic Number Theory — J. S. Milne. Direct open PDF
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You can use Dedekind-domain and field-extension arguments; class field theory and cohomology may require more background.
Research reading: additional tools
Choose according to the particular text, not as an obligation to complete every foundation again.