Abstract Algebra Prerequisites: Number Theory, Set Theory and Linear Algebra
Abstract Algebra Prerequisites: Number Theory, Set Theory and Linear Algebra: core definitions, structural results and verification methods in abstract algebr
How the topic fits together
Elementary Number Theory
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Set Theory
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Linear Algebra
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Core definitions and structural results
The following cards retain the mathematical content of the source while condensing long proofs into verification strategies. Numerical examples are treated as examples, not universal requirements.
If d is the greatest common divisor of a and b, then there are integers s and t such
If d is the greatest common divisor of a and b, then there are integers s and t such that sa + tb = d. In particular, if a and b are relatively prime, there are integers s and t such that sa + tb = 1.
Proof / verification strategy: Assume a non-trivial factorisation or divisibility relation and use degree, content, ideal or prime-divisibility constraints to force one factor to be a unit or to obtain a contradiction.
If a prime p divides a product a1 · · · an of integers, then p divides at least one ai
If a prime p divides a product a1 · · · an of integers, then p divides at least one ai
Proof / verification strategy: Assume a non-trivial factorisation or divisibility relation and use degree, content, ideal or prime-divisibility constraints to force one factor to be a unit or to obtain a contradiction.
Unique Factorization Theorem If a is an integer, not 0 or ±1, then
Unique Factorization Theorem If a is an integer, not 0 or ±1, then (1) a can be written as a product p1 · · · pn of primes. (2) If a = p1 · · · pn = q1 · · · qm, where the pi and qj are prime, then n = m and, after renumbering, pi = ±qi for all i.
Proof / verification strategy: Assume a non-trivial factorisation or divisibility relation and use degree, content, ideal or prime-divisibility constraints to force one factor to be a unit or to obtain a contradiction.
The Integers Modulo m If a and b are integers and m is a positive integer ≥2,
The Integers Modulo m If a and b are integers and m is a positive integer ≥2, we write a ≡b mod m, and say that a is congruent to b modulo m, if a −b is divisible by m. Congruence modulo m is an equivalence relation, and the resulting equivalence classes are called residue classes mod m. Residue classes can be added, subtracted and multiplied consistently by choosing a representative from each class, performing the appropriate operation, and calculating the residue class of the result. The collection Zm of residue classes mod m forms a commutative ring under addition and multiplication.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Chinese Remainder Theorem If m1, . . . , mr are relatively prime in pairs, then the
Chinese Remainder Theorem If m1, . . . , mr are relatively prime in pairs, then the system of simultaneous equations x ≡bj mod mj, j = 1, . . . , r, has a solution for arbitrary integers bj. The set of solutions forms a single residue class mod m=m1 · · · mr, so that there is a unique solution mod m. This result can be derived from the abstract form of the Chinese remainder theorem; see Section 2.3.
Proof / verification strategy: Assume a non-trivial factorisation or divisibility relation and use degree, content, ideal or prime-divisibility constraints to force one factor to be a unit or to obtain a contradiction.
Euler’s Theorem
Euler’s Theorem The Euler phi function is defined by ϕ(n) = the number of integers in {1, . . . , n} that are relatively prime to n. For an explicit formula for ϕ(n), see
Proof / verification strategy: Assume a non-trivial factorisation or divisibility relation and use degree, content, ideal or prime-divisibility constraints to force one factor to be a unit or to obtain a contradiction.
Fermat’s Little Theorem If a is any integer and p is a prime not dividing a, then
Fermat’s Little Theorem If a is any integer and p is a prime not dividing a, then ap−1 ≡1 mod p. Thus for any integer a and prime p, whether or not p divides a, one has ap ≡a mod p. For proofs of (0.1.7) and (0.1.8), see (1.3.4). 0.2 Set Theory
Proof / verification strategy: Assume a non-trivial factorisation or divisibility relation and use degree, content, ideal or prime-divisibility constraints to force one factor to be a unit or to obtain a contradiction.
A partial ordering on a set S is a relation on S that is reflexive (x ≤x for all x ∈S),
A partial ordering on a set S is a relation on S that is reflexive (x ≤x for all x ∈S), antisymmetric (x ≤y and y ≤x implies x = y), and transitive (x ≤y and y ≤z implies x ≤z). If for all x, y ∈S, either x ≤y or y ≤x, the ordering is total.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
A well-ordering on S is a partial ordering such that every nonempty subset A of S
A well-ordering on S is a partial ordering such that every nonempty subset A of S has a smallest element a. (Thus a ≤b for every b ∈A).
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Well-Ordering Principle Every set can be well-ordered.
Well-Ordering Principle Every set can be well-ordered.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Maximum Principle If T is any chain (totally ordered subset) of a partially ordered
Maximum Principle If T is any chain (totally ordered subset) of a partially ordered set S, then T is contained in a maximal chain M. (Maximal means that M is not properly contained in a larger chain.)
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Zorn’s Lemma If S is a nonempty partially ordered set such that every chain of S
Zorn’s Lemma If S is a nonempty partially ordered set such that every chain of S has an upper bound in S, then S has a maximal element. (The element x is an upper bound of the set A if a ≤x for every a ∈A. Note that x need not belong to A, but in the statement of Zorn’s lemma, we require that if A is a chain of S, then A has an upper bound that actually belongs to S.)
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Axiom of Choice Given any family of nonempty sets Si, i ∈I, we can choose an
Axiom of Choice Given any family of nonempty sets Si, i ∈I, one can choose an element of each Si. Formally, there is a function f whose domain is I such that f(i) ∈Si for all i ∈I. The well-ordering principle, the maximum principle, Zorn’s lemma, and the axiom of choice are equivalent in the sense that if any one of these statements is added to the basic axioms of set theory, all the others can be proved. The statements themselves cannot be proved from the basic axioms.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
We say that the size of the set A is less than or equal to the size of B (notation
Call the size of the set A is less than or equal to the size of B (notation A ≤s B) if there is an injective map from A to B. Call A and B have the same size (A =s B) if there is a bijection between A and B.
Proof / verification strategy: Use the universal or lifting property that defines the construction, then check naturality and the effect on exact sequences.
Schr¨oder-Bernstein Theorem If A ≤s B and B ≤s A, then A =s B. (This can
Schr¨oder-Bernstein Theorem If A ≤s B and B ≤s A, then A =s B. (This can be proved without the axiom of choice.)
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Using (0.2.9), one can show that if sets of the same size are called equivalent, then
Using (0.2.9), one can show that if sets of the same size are called equivalent, then ≤s on equivalence classes is a partial ordering. It follows with the aid of Zorn’s lemma that
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
For any set A, we can always produce a set of greater cardinality, namely the power
For any set A, one can always produce a set of greater cardinality, namely the power set 2A, that is, the collection of all subsets of A.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Define addition and multiplication of cardinal numbers by |A| + |B| = |A ∪B| and
Define addition and multiplication of cardinal numbers by |A| + |B| = |A ∪B| and |A||B| = |A × B|. In defining addition, we assume that A and B are disjoint. (They can always be disjointized by replacing a ∈A by (a, 0) and b ∈B by (b, 1).)
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
If ℵ0 is the cardinal number of a countably infinite set, then ℵ0 + ℵ0 = ℵ0ℵ0 = ℵ0.
If ℵ0 is the cardinal number of a countably infinite set, then ℵ0 + ℵ0 = ℵ0ℵ0 = ℵ0. More generally, (a) If α and β are cardinals, with α ≤β and β infinite, then α + β = β. (b) If α ̸= 0 (i.e., α is nonempty), α ≤β and β is infinite, then αβ = β.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
If A is an infinite set, then A and the set of all finite subsets of A have the same
If A is an infinite set, then A and the set of all finite subsets of A have the same cardinality. 0.3 Linear Algebra It is not feasible to list all results presented in an undergraduate course in linear algebra. Instead, here is a list of topics that are covered in a typical course. 1. Sums, products, transposes, inverses of matrices; symmetric matrices. 2. Elementary row and column operations; reduction to echelon form. 3.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Quick-reference relationships
Problem-solving workflow
Identify the ambient domain and extension
State the base ring or field and the integral elements under discussion.
Translate arithmetic into ideals or linear maps
Norms, traces, discriminants and ideal factorisation encode arithmetic structurally.
Track divisibility and integrality
Do not confuse being algebraic with being integral, or element factorisation with ideal factorisation.
Use local information where appropriate
Valuations and p-adic ideas measure divisibility using a topology different from the usual absolute value.
Check finiteness hypotheses
Finite generation, Noetherian conditions and finite extensions are frequently essential.
Verify with quadratic or integer examples
Use small extensions and ideals to test signs, degrees and multiplicities.
Worked-solution emphasis from the supplied source
The source exercises reinforce definition checking, construction of examples and direct use of the formal results in this article.
Common mistakes and boundary conditions
- Skipping a hypothesis because it seems automatic.
- Using notation before defining the ambient structure.
- Treating an illustrative calculation as a universal result.
- Relying on intuition without verifying closure, well-definedness or exactness.
Verification checklist
- State the ambient algebraic structure and operation before applying a theorem.
- Record every hypothesis that controls the result: finiteness, commutativity, normality, primality, separability, exactness or other section-specific conditions.
- Distinguish a definition from a theorem that follows from it.
- Check whether a map is well-defined before using its kernel, image, inverse or induced map.
- Use a concrete example only as a check; do not promote an illustrative value or pattern to a universal rule.
- When a quotient, localisation or extension is constructed, identify the canonical map and what becomes equal, invertible or fixed.
Source coverage map
| Source section | Subject | PDF pages analysed |
|---|---|---|
| 0.1 | Elementary Number Theory | 7–7 |
| 0.2 | Set Theory | 8–8 |
| 0.3 | Linear Algebra | 9–9 |
Related Mathematics pages
Source note: synthesised from the supplied abstract-algebra PDF. The complete 298-page file, including diagrams and worked solutions, was reviewed. Source-identifying author and bibliographic personal details are intentionally omitted. Formal proofs are condensed; the page does not claim requirements or values not supported by the supplied mathematics.