Primordial element (algebra)

Last updated May 05, 2024

In algebra, a primordial element is a particular kind of a vector in a vector space.

Definition

Let $V$ be a vector space over a field $\mathbb {F}$ and let $\left(e_{i}\right)_{i\in I}$ be an $I$ -indexed basis of vectors for $V.$ By the definition of a basis, every vector $v\in V$ can be expressed uniquely as

v=\sum _{i\in I}a_{i}(v)e_{i}

for some $I$ -indexed family of scalars $\left(a_{i}\right)_{i\in I}$ where all but finitely many $a_{i}$ are zero. Let

I(v)=\left\{i\in I:a_{i}(v)\neq 0\right\}

denote the set of all indices for which the expression of $v$ has a nonzero coefficient. Given a subspace $W$ of $V,$ a nonzero vector $p\in W$ is said to be primordial if it has both of the following two properties:^[1]

$I(p)$ is minimal among the sets $I(w),$ where $0\neq w\in W,$ and
$a_{i}(p)=1$ for some index $i.$

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References

↑ Milne, J., Class field theory course notes, updated March 23, 2013, Ch IV, §2.

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[1] Milne, J., Class field theory course notes, updated March 23, 2013, Ch IV, §2.

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v t e Linear algebra
Outline Glossary
Basic concepts	Scalar Vector Vector space Scalar multiplication Vector projection Linear span Linear map Linear projection Linear independence Linear combination Basis Change of basis Row and column vectors Row and column spaces Kernel Eigenvalues and eigenvectors Transpose Linear equations
Matrices	Block Decomposition Invertible Minor Multiplication Rank Transformation Cramer's rule Gaussian elimination
Bilinear	Orthogonality Dot product Hadamard product Inner product space Outer product Kronecker product Gram–Schmidt process
Multilinear algebra	Determinant Cross product Triple product Seven-dimensional cross product Geometric algebra Exterior algebra Bivector Multivector Tensor Outermorphism
Vector space constructions	Dual Direct sum Function space Quotient Subspace Tensor product
Numerical	Floating-point Numerical stability Basic Linear Algebra Subprograms Sparse matrix Comparison of linear algebra libraries
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