= Linear independence of distinct field embeddings
= Artin independence theorem
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{synonym}
Distinct field embeddings $\sigma_1,\ldots,\sigma_n:L\to\Omega$ are linearly independent over $\Omega$ as functions on $L$. Thus a nontrivial linear combination $\sum_i a_i\sigma_i$ cannot vanish on every element of $L$.
For the proof, choose a vanishing relation with the fewest nonzero coefficients and normalize one coefficient to one. If $\sigma_1\ne\sigma_m$, choose $y$ with $\sigma_1(y)\ne\sigma_m(y)$. Evaluating at $xy$ and subtracting $\sigma_m(y)$ times the relation evaluated at $x$ eliminates its $m$th term but leaves a nonzero first coefficient, contradicting minimality.
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