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Introduction to algebra
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I · Algebra basics
Chapter I.1
Origins of algebra
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On this page
I.1.1 · Brief algebra
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I.1.2 · Some model problems
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I.1.3 · Systems of linear equations. First steps
I.1.4 · Determinants of low orders
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I.1.5 · Sets and mappings
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I.1.6 · Equivalence relations. Factorization of mappings
I.1.7 · The principle of mathematical induction
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I.1.8 · Permutations
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I.1.9 · Arithmetic of integers
Section I.1.1
Brief algebra
Section I.1.2
Some model problems
Subsection I.1.2.1
The problem of solvability of equations in radicals
Subsection I.1.2.2
The problem of states of a polyatomic molecule
Subsection I.1.2.3
Message encoding problem
Subsection I.1.2.4
The problem of a heated plate
Section I.1.3
Systems of linear equations. First steps
Subsection I.1.3.1
Terminology
Subsection I.1.3.2
Equivalence of linear systems
Subsection I.1.3.3
Reduction to row echelon form
Subsection I.1.3.4
Study of a system of linear equations
Subsection I.1.3.5
Separate remarks and examples
Section I.1.4
Determinants of low orders
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Section I.1.5
Sets and mappings
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Subsection I.1.5.1
Sets
Subsection I.1.5.2
Maps
Section I.1.6
Equivalence relations. Factorization of mappings
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Subsection I.1.6.1
Binary relations
Subsection I.1.6.2
Equivalence relation
Subsection I.1.6.3
Factorization of mappings
Subsection I.1.6.4
Ordered sets
Section I.1.7
The principle of mathematical induction
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Section I.1.8
Permutations
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Subsection I.1.8.1
Standard notation for a permutation
Subsection I.1.8.2
Cycle structure of a permutation
Subsection I.1.8.3
Sign of a permutation
Subsection I.1.8.4
The action of
S
n
S_n
S
n
on functions
Section I.1.9
Arithmetic of integers
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Subsection I.1.9.1
The Fundamental Theorem of Arithmetic
Subsection I.1.9.2
GCD and LCM in
Z
\mathbb{Z}
Z
Subsection I.1.9.3
Division algorithm in
Z
\mathbb{Z}
Z