Weierstrass Nullstellensatz

In mathematics, the Weierstrass Nullstellensatz is a version of the intermediate value theorem over a real closed field. It says:

Given a polynomial $f$ in one variable with coefficients in a real closed field F and $a<b$ in $F$, if $f(a)<0<f(b)$, then there exists a $c$ in $F$ such that $a<c<b$ and $f(c)=0$.

Proof

Since F is real-closed, F(i) is algebraically closed, hence f(x) can be written as $u\prod _i(x-{\alpha }_i)$, where $u\in F$ is the leading coefficient and ${\alpha }_j\in F(i)$ are the roots of f. Since each nonreal root ${\alpha }_j=a_j+i{b}_j$ can be paired with its conjugate ${\bar{\alpha }}_j=a_j-i{b}_j$ (which is also a root of f), we see that f can be factored in F[x] as a product of linear polynomials and polynomials of the form $(x-{\alpha }_j)(x-{\bar{\alpha }}_j)=(x-a_j{)}^2+b_j^2$, $b_j\ne 0$.

If f changes sign between a and b, one of these factors must change sign. But $(x-a_j{)}^2+b_j^2$ is strictly positive for all x in any formally real field, hence one of the linear factors $x-{\alpha }_j$, ${\alpha }_j\in F$, must change sign between a and b; i.e., the root ${\alpha }_j$ of f satisfies $a<{\alpha }_j<b$.