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Primordial element (algebra)

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

Definition

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Let be a vector space over a field and let be an -indexed basis of vectors for By the definition of a basis, every vector can be expressed uniquely as for some -indexed family of scalars where all but finitely many are zero. Let denote the set of all indices for which the expression of has a nonzero coefficient. Given a subspace of a nonzero vector is said to be primordial if it has both of the following two properties:[1]

  1. is minimal among the sets where and
  2. for some index

References

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