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For example, does the "minimal set" $\{1,\sqrt 2,\sqrt 3,\sqrt 6 \}$ form a basis over $\mathbb{Q}$?

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    Thank you for noting that Rankeya2012-03-28

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If $S$ is a set of vectors linearly independent over a field $F$, then $S$ is a basis over $F$ for the span of $S$ (which is an $F$-vector space).