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Algebra
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Mathematics
Area of mathematics
OurBigBook.com
words: 8k
articles: 263
Not to be confused with
algebra over a field
, which is a particular
algebraic structure
studied within algebra.
Table of contents
8k
263
Abstract algebra
Algebra
10
Algebraic structure
Algebra
175
7
Commutator
Algebraic structure
Identity element
Algebraic structure
3
2
Inverse element
Identity element
3
1
Invertible
Inverse element
Order
(algebra)
Algebraic structure
131
2
Degree
(algebra)
Order
86
Finite algebraic structure
Order
1
Linear algebra
Algebra
3k
127
Linear function
Linear algebra
Linear map
(linear operator)
Linear algebra
1k
27
Form
(mathematics)
Linear map
Linear form
Linear map
174
3
Matrix representation of a linear form
Linear form
Dual space
(
V
∗
)
Linear form
1
Dual vector
(
e
i
)
Dual space
Linear operator
Linear map
1
Adjoint operator
(
A
†
)
Linear operator
Self-adjoint operator
Linear map
Multilinear map
Linear map
570
18
Bilinear map
Multilinear map
Bilinear form
(
B
(
x
,
y
)
)
Multilinear map
157
2
Matrix representation of a bilinear form
Bilinear form
1
Effect of a change of basis on the matrix of a bilinear form
(
B
2
=
C
T
B
C
)
Matrix representation of a bilinear form
Multilinear form
Multilinear map
Symmetric bilinear map
Multilinear map
80
9
Symmetric bilinear form
Symmetric bilinear map
1
Matrix representation of a symmetric bilinear form
Symmetric bilinear form
Hermitian form
Symmetric bilinear map
1
Matrix representation of a Hermitian form
Hermitian form
Quadratic form
Symmetric bilinear map
Positive definite symmetric bilinear form
Symmetric bilinear map
1
Matrix representation of a positive definite symmetric bilinear form
Positive definite symmetric bilinear form
Skew-symmetric bilinear map
Symmetric bilinear map
Skew-symmetric bilinear form
Symmetric bilinear map
Symmetric multilinear map
Multilinear map
1
Antisymmetric multilinear map
Symmetric multilinear map
Alternating multilinear map
Multilinear map
Dot product
Linear algebra
3
Orthogonality
Dot product
1
Orthonormality
Orthogonality
Angle
Dot product
Cross product
(
x
×
y
)
Linear algebra
1
Jacobi identity
Cross product
Index picking function
Linear algebra
70
3
Kronecker delta
(
δ
i
j
)
Index picking function
Levi-Civita symbol
(
ε
)
Index picking function
1
Levi-Civita symbol as a tensor
Levi-Civita symbol
Projection
(mathematics)
Linear algebra
Matrix
Linear algebra
984
55
Matrix operation
Matrix
42
6
Determinant
(
d
e
t
)
Matrix operation
Matrix inverse
(
M
−
1
)
Matrix operation
1
Invertible matrix
Matrix inverse
Transpose
(
M
T
)
Matrix operation
2
Transpose of a matrix multiplication
Transpose
Inverse of the transpose
Transpose
Matrix multiplication
Matrix
102
9
System of linear equations
Matrix multiplication
76
4
Application of systems of linear equations
System of linear equations
System of linear equations algorithm
System of linear equations
2
LINPACK benchmarks
System of linear equations algorithm
Conjugate gradient method
System of linear equations algorithm
Application of matrix multiplication
Matrix multiplication
Matrix multiplication algorithm
Matrix multiplication
1
General Matrix Matrix Multipliation
Matrix multiplication algorithm
Matrix decomposition
Matrix multiplication
Eigenvalues and eigenvectors
Matrix
501
19
Applications of eigenvalues and eigenvectors
Eigenvalues and eigenvectors
Characteristic polynomial
Eigenvalues and eigenvectors
Eigenvalue
Eigenvalues and eigenvectors
49
2
Spectrum
(functional analysis)
Eigenvalue
1
Continuous spectrum
(functional analysis)
Spectrum
Eigendecomposition of a matrix
Eigenvalues and eigenvectors
63
7
Eigendecomposition of a real symmetric matrix
Eigendecomposition of a matrix
Sylvester's law of inertia
Eigendecomposition of a matrix
41
5
Congruent matrix
Sylvester's law of inertia
1
Matrix congruence can be seen as the change of basis of a bilinear form
Congruent matrix
Matrix similarity
Sylvester's law of inertia
Metric signature
Sylvester's law of inertia
1
Metric signature matrix
Metric signature
Eigenvector
Eigenvalues and eigenvectors
Eigenvectors and eigenvalues of the identity matrix
Eigenvalues and eigenvectors
Spectral theorem
Eigenvalues and eigenvectors
40
3
Hermitian matrix
(complex analogue of symmetric matrix)
Spectral theorem
2
Hermitian operator
Hermitian matrix
1
Riesz representation theorem
Hermitian operator
Kronecker product
Matrix
Named matrix
Matrix
100
16
Dense and sparse matrices
Named matrix
2
Dense matrix
Dense and sparse matrices
Sparse matrix
Dense and sparse matrices
Diagonal matrix
Named matrix
2
Scalar matrix
(
Z
(
V
)
)
Diagonal matrix
1
Identity matrix
(
I
n
)
Scalar matrix
Square matrix
Named matrix
1
Matrix ring
Square matrix
Orthogonal matrix
Named matrix
1
Unitary matrix
Orthogonal matrix
Triangular matrix
Named matrix
Symmetric matrix
Named matrix
18
4
Definite matrix
Symmetric matrix
1
Positive definite matrix
Definite matrix
Skew-symmetric matrix
Symmetric matrix
1
Skew-symmetric form
Skew-symmetric matrix
Vector space
Linear algebra
226
10
Basis
(linear algebra)
Vector space
175
4
Change of basis
Basis
2
Change of basis matrix
Change of basis
Change of basis between symmetric matrices
Change of basis
Linear independence
Basis
Classification of vector spaces
Vector space
Underlying field of a vector space
Vector space
Tensor product
(
⊗
)
Vector space
Vector
(mathematics)
Vector space
1
Scalar
(mathematics)
Vector
Tensor
Linear algebra
487
17
A linear map is a (1,1) tensor
Tensor
Tensor space
(
T
(
m
,
n
)
)
Tensor
1
Order of a tensor
Tensor space
Einstein notation
Tensor
285
13
Raised and lowered indices
Einstein notation
3
Raised index
Raised and lowered indices
Lowered index
Raised and lowered indices
Raising and lowering indices
Raised and lowered indices
Implicit metric signature in Einstein notation
Einstein notation
Einstein notation for partial derivatives
Einstein notation
88
4
Divergence in Einstein notation
(
∂
i
)
Einstein notation for partial derivatives
Laplacian in Einstein notation
(
∂
i
∂
i
)
Einstein notation for partial derivatives
30
2
D'alembert operator in Einstein notation
(
∂
i
∂
i
)
Laplacian in Einstein notation
1
Klein-Gordon equation in Einstein notation
D'alembert operator in Einstein notation
Covariance and contravariance of vectors
Einstein notation
2
Covariant vector
Covariance and contravariance of vectors
Contravariant vector
Covariance and contravariance of vectors
Linear algebra bibliography
Linear algebra
1
Interactive Linear Algebra by Margalit and Rabinoff
Linear algebra bibliography
Group
Algebra
3k
98
Center
(group theory)
Group
Group axiom
Group
10
3
Commutative property
Group axiom
2
Abelian group
Commutative property
1
Non-commutative
Abelian group
Symmetry
Group
1
Symmetry breaking
Symmetry
Important mathematical group
Group
301
19
Important discrete mathematical group
Important mathematical group
188
18
Cyclic group
(
C
n
)
Important discrete mathematical group
The direct product of two cyclic groups of coprime order is another cyclic group
Important discrete mathematical group
Permutation
Important discrete mathematical group
140
11
Cycle notation
Permutation
Parity of a permutation
Permutation
2
Odd permutation
Parity of a permutation
Even permutation
Parity of a permutation
Permutation group
Permutation
21
6
Stabilizer
(group)
Permutation group
Symmetric group
Permutation group
1
All groups are isomorphic to a subgroup of the symmetric group
Symmetric group
Alternating group
Permutation group
9
2
Alternating group of degree 5
Alternating group
1
The alternating groups of degree 5 or greater are simple
Alternating group of degree 5
Dihedral group
(
D
n
)
Important discrete mathematical group
Wallpaper group
Important discrete mathematical group
Space group
Important discrete mathematical group
Klein four-group
Important discrete mathematical group
Finite group
Group
668
28
Classification of finite groups
Finite group
494
27
List of finite groups
Classification of finite groups
1
GroupNames
List of finite groups
Classification of finite simple groups
Classification of finite groups
356
23
Group of Lie type
Classification of finite simple groups
30
2
Chevalley group
Group of Lie type
1
Chevalley groups
A
n
(
q
)
Chevalley group
Sporadic group
Classification of finite simple groups
125
17
Mathieu group
Sporadic group
93
13
k-transitive group
Mathieu group
82
7
Classification of k-transitive groups
k-transitive group
6
2-transitive group
Classification of k-transitive groups
1
Classification of 2-transitive groups
2-transitive group
Classification of 3-transitive groups
Classification of k-transitive groups
Classification of 4-transitive groups
Classification of k-transitive groups
Classification of 5-transitive groups
Classification of k-transitive groups
Classification of 6-transitive groups
Classification of k-transitive groups
Mathieu group
M
1
1
Mathieu group
Mathieu group
M
1
2
Mathieu group
Mathieu group
M
2
2
Mathieu group
Mathieu group
M
2
3
Mathieu group
Mathieu group
M
2
4
Mathieu group
Janko group
Sporadic group
Monster group
Sporadic group
1
Monstrous moonshine
Monster group
Jordan-Holder Theorem
Classification of finite simple groups
Composition series
Classification of finite simple groups
Group extension problem
Classification of finite groups
Group operation
Group
Group isomorphism
Group
7
4
Isomorphism
Group isomorphism
Group homomorphism
Group isomorphism
2
Fundamental theorem on homomorphisms
Group homomorphism
Kernel
(algebra)
Group homomorphism
Generating set of a group
Group
112
6
Finitely generated group
Generating set of a group
Rank of a group
Generating set of a group
Cayley graph
Generating set of a group
1
Cycle graph
(algebra)
Cayley graph
Cycle of an element of a group
Generating set of a group
1
Order of an element of a group
Cycle of an element of a group
Direct product of groups
(
G
×
H
)
Group
2
Product of group subsets
Direct product of groups
Semidirect product
(
N
⋊
H
)
Direct product of groups
Subgroup
Group
457
7
Subgroup generated by a group
Subgroup
Quotient group
Subgroup
45
5
Subquotient
Quotient group
Relationship between the quotient group and direct products
Quotient group
Normal subgroup
Quotient group
11
2
Simple group
Normal subgroup
1
How to show that a group is simple
Simple group
Ring
(mathematics)
Group
750
17
Commutative ring
Ring
Division ring
Ring
Finite ring
Ring
1
Classification of finite rings
Finite ring
Field
(mathematics)
Ring
506
12
Multiplicative inverse
Field
Distributive property
Field
Finite field
(
G
F
(
n
)
)
Field
4
Classification of finite fields
Finite field
Finite field of non-prime order
Finite field
GF(2)
Finite field
GF(4)
Finite field
Quadratically closed field
Field
Vector field
Field
14
3
Algebra over a field
Vector field
11
2
Division algebra
Algebra over a field
1
Frobenius theorem
(real division algebras)
Division algebra
Associative property
Algebra
Algebraic geometry
Algebra
1k
25
The beauty of alebraic geometry
Algebraic geometry
10
Algebraic curve
Algebraic geometry
Elliptic curve
Algebraic geometry
1k
22
Elliptic curve group
Elliptic curve
224
2
Elliptic curve point addition
Elliptic curve group
201
Elliptic curve point multiplication
Elliptic curve group
Domain of an elliptic curve
Elliptic curve
804
16
Not every
x
belongs to the elliptic curve over a non quadratically closed field
Domain of an elliptic curve
229
1
Number of elements of an elliptic curve
Not every
x
belongs to the elliptic curve over a non quadratically closed field
Elliptic curve over the real numbers
(
E
(
F
)
)
Domain of an elliptic curve
Elliptic curve over the rational numbers
(
E
(
Q
)
)
Domain of an elliptic curve
568
9
Number of elements of an elliptic curve over the rational numbers
Elliptic curve over the rational numbers
153
3
Mordell's theorem
(1922)
Number of elements of an elliptic curve over the rational numbers
140
2
Rank of an elliptic curve over the rational numbers
(
r
)
Mordell's theorem
106
1
Largest known ranks of an elliptic curve over the rational numbers
Rank of an elliptic curve over the rational numbers
29
Reduction of an elliptic curve over the rational numbers to an elliptic curve over a finite field mod p
(Reduction of an elliptic curve from
E
(
Q
)
to
E
(
F
p
)
m
o
d
p
)
Elliptic curve over the rational numbers
106
Birch and Swinnerton-Dyer conjecture
(1965, BSD conjecture)
Elliptic curve over the rational numbers
309
3
BSD conjecture bibliography
Birch and Swinnerton-Dyer conjecture
53
2
Birch and Swinnerton-Dyer conjecture in two minutes by Ciro Santilli
(2023)
BSD conjecture bibliography
38
Notes on Elliptic Curves (II) by BSD
(1965)
BSD conjecture bibliography
15
Elliptic curve over a finite field
(
E
(
F
p
)
)
Domain of an elliptic curve
7
2
Number of elements of an elliptic curve over a finite field
Elliptic curve over a finite field
1
Schoof's algorithm
Number of elements of an elliptic curve over a finite field
Elliptic curve bibliography
Elliptic curve
1
Elliptic curve university course
Elliptic curve bibliography
Ancestors
(3)
Area of mathematics
Mathematics
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(1)
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