b
= b
{scope}
For a <splitting field> $L$ of $f$ over $K$, define $\operatorname{Gal}(f)=\operatorname{Aut}_K(L)$. Each <automorphism> permutes the distinct roots, since it fixes the coefficients. An <automorphism> fixing every root fixes the <field> they generate, so this action is faithful. If there are $r\le n$ distinct roots, it gives an embedding into $S_r$, and hence into $S_n$ by fixing $n-r$ additional symbols.