Primordial element (algebra)

In algebra, a primordial element is a particular kind of a vector in a vector space. Let V be a vector space over a field k and fix a basis for V of vectors $e_i$ for $i \in I$. By the definition of a basis, every vector v in V can be expressed uniquely as

$v = \sum_{i \in I} a_i(v) e_i.$

Define $I(v) = \{ i \in I \mid a_i(v) \ne 0 \}$, the set of indices for which the expression of v has a nonzero coefficient. Given a subspace W of V, a nonzero vector w in W is said to be "primordial" if it has the following two properties:^[1]

1. $I(w)$ is minimal among the sets $I(w')$, $0 \ne w' \in W$ and
2. $a_i(w) = 1$ for some i

References

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