abstract algebra抽象代数ちゅうしょうだいすう portal

date2026-06-06document_iddoc_cc782fa623f5c4e76b366dd1c064e76cdescription抽象代数を、演算・同値関係・商構造・群・環・体・準同型をつなぐ章として学ぶためのポータル。prerequisites[命題/めいだい]・[述語/じゅつご]と[量化/りょうか] / [集合/しゅうごう]の[基本/きほん] / [写像/しゃぞう]の[基本/きほん]type講義content_typelecturestatusactive

mathabstract-algebraundergraduatelecture

abstract algebra抽象代数ちゅうしょうだいすう studies the form of operations演算えんざん carried by numbers, transformations, and symmetries対称性たいしょうせい, rather than studying numbers only as individual objects. The first important point is not to memorize groups群ぐん, rings環かん, and fields体たい as isolated names. They are languages for organizing which operations are allowed, what those operations preserve, and what information they forget.

The purpose of abstraction is not to throw away concrete examples. It is to extract the structure構造こうぞう shared by concrete examples so that the same argument can be transported to other objects.

1Viewpoints needed first

abstract algebra抽象代数ちゅうしょうだいすう rests on discrete mathematics. Review the following topics before starting.

data/lecture/math/discrete-math/propositions-predicates-and-quantifiers.lecture.n.md data/lecture/math/discrete-math/equivalence-relations-and-partitions.lecture.n.md data/lecture/math/discrete-math/quotient-sets-and-canonical-projections.lecture.n.md data/lecture/math/discrete-math/map-basics.lecture.n.md

The next two prerequisites are needed later and do not have to be completed before starting. Study the Euclidean algorithm before “Congruences and modular arithmetic,” and vector spaces and bases before “Introduction to finite fields.”

data/lecture/math/algebra/euclidean-algorithm-and-linear-diophantine-equations.lecture.n.md data/lecture/math/linear-algebra/vector-spaces-and-bases.lecture.n.md

2Reading order順序じゅんじょ

First, look at operations and structures.

data/lecture/math/abstract-algebra/introduction-to-algebraic-structures.lecture.n.md data/lecture/math/abstract-algebra/binary-operations-and-closure.lecture.n.md data/lecture/math/abstract-algebra/semigroups-monoids-and-groups.lecture.n.md

Next, learn how to treat different elements as the same. This is the prerequisite for quotient groups商群しょうぐん and quotient rings商環しょうかん.

data/lecture/math/abstract-algebra/equivalence-relations-and-cosets.lecture.n.md data/lecture/math/abstract-algebra/congruences-and-modular-arithmetic.lecture.n.md

After that, study groups群ぐん.

data/lecture/math/abstract-algebra/group-basics.lecture.n.md data/lecture/math/abstract-algebra/subgroups-and-generators.lecture.n.md data/lecture/math/abstract-algebra/cosets-and-lagrange-theorem.lecture.n.md data/lecture/math/abstract-algebra/normal-subgroups-and-quotient-groups.lecture.n.md data/lecture/math/abstract-algebra/group-homomorphisms-and-isomorphisms.lecture.n.md data/lecture/math/abstract-algebra/group-actions-and-symmetry.lecture.n.md

Next, move to rings環かん and fields体たい, which are structures with two operations.

data/lecture/math/abstract-algebra/ring-basics.lecture.n.md data/lecture/math/abstract-algebra/ideals-and-quotient-rings.lecture.n.md data/lecture/math/abstract-algebra/integral-domains-zero-divisors-and-polynomial-rings.lecture.n.md data/lecture/math/abstract-algebra/field-basics.lecture.n.md data/lecture/math/abstract-algebra/introduction-to-finite-fields.lecture.n.md

Finally, study homomorphisms and the first isomorphism theorem across groups and rings.

data/lecture/math/abstract-algebra/homomorphism-basics.lecture.n.md data/lecture/math/abstract-algebra/homomorphism-theorems-overview.lecture.n.md

3What changes and what is preserved

Viewpoint	What changes	What should be preserved
equivalence relation同値関係どうちかんけい	Individual representatives	The classification of belonging to the same class
quotient structure商構造しょうこうぞう	Elements are grouped into classes	Operations do not depend on representatives
homomorphism準同型じゅんどうけい	The representation of an object	The operational structure
isomorphism同型どうけい	The names of elements	All algebraic relations
group action群作用ぐんさよう	A group is viewed as transformations	The structure of symmetry

This table is a guide for reading all of abstract algebra抽象代数ちゅうしょうだいすう. Whenever you meet a definition定義ていぎ, always ask: what does this definition定義ていぎ change, and what is it designed not to change?

4Exercise links

The exercises are also listed in study order. Some exercise files cover several lectures, so check each file's "Corresponding lectures" section and any problem-level sequencing notes.

5Summary

abstract algebra抽象代数ちゅうしょうだいすう is a language for comparing objects that carry operations. A group群ぐん has one operation, while a ring環かん has addition and multiplication. A field体たい is a commutative ring with $0 \neq 1$ in which every nonzero element has a multiplicative inverse. Through homomorphisms準同型じゅんどうけい and quotient structures商構造しょうこうぞう, concrete calculation and abstract structure are connected.