Let $E$ be a finite extension over $K$. If $E = K(a_1,a_2,\cdots,a_n)$ and each $a_i$ is separable over $K$ then can we say that any linear combination of $\{a_1,a_2,\cdots,a_n\}$ is also separable over $K$?
Any linear combination of $\{a_1,a_2,\cdots,a_n\}$ is also separable over $K$
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linear-algebra
extension-field
separable-spaces
1 Answers
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The set of separable elements over a field form a field in itself, and in particular linear combinations. See for example here.