Determinant trick
= Determinant trick
{wiki}
If a finitely generated faithful <module> $M$ over a <commutative ring> $R$ satisfies $xM\subseteq IM$, the <determinant> of a <matrix> presenting these relations gives a monic equation
$$
x^n+a_{n-1}x^{n-1}+\cdots+a_0=0,
\qquad a_i\in I^{n-i}.
$$
In particular, an element of an extension field that preserves a nonzero finitely generated <ideal> is integral over $R$.