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 $I(v)=\{i\in I\mid a_{i}(v)\neq 0\}$ , 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]

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

References

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

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