Matroid Induced by Algebraic Independence is Matroid

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Theorem

Let $L / K$ be a field extension.

Let $S \subseteq L$ be a finite subset of $L$.

Let $\struct {S, \mathscr I}$ be the matroid induced by algebraic independence over $K$ on $S$.

That is, $\mathscr I$ is the set of algebraically independent subsets of $S$.


Then $\struct {S, \mathscr I}$ is a matroid.

Proof

It needs to be shown that $\mathscr I$ satisfies the matroid axioms $(\text I 1)$, $(\text I 2)$ and $(\text I 3)$.

Template:Matroid-axiom


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Matroid Axiom $(\text I 2)$


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Matroid Axiom $(\text I 3)$


This theorem requires a proof.
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Sources

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