Primordial element (algebra)

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 𝔽 and let (ei)i∈I be an I-indexed basis of vectors for V. By the definition of a basis, every vector v∈V can be expressed uniquely as v=∑i∈Iai(v)ei for some I-indexed family of scalars (ai)i∈I where all but finitely many ai are zero. Let I(v)={i∈I:ai(v)≠0} 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∈W is said to be primordial if it has both of the following two properties:[1]

1. I(p) is minimal among the sets I(w), where 0≠w∈W, and
2. ai(p)=1 for some index i.

## References

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

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