Weierstrass Nullstellensatz

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

Proof

Since F is real-closed, F is algebraically closed, hence f can be written as, where is the leading coefficient and are the roots of f. Since each nonreal root can be paired with its conjugate, we see that f can be factored in F as a product of linear polynomials and polynomials of the form,.
If f changes sign between a and b, one of these factors must change sign. But is strictly positive for all x in any formally real field, hence one of the linear factors,, must change sign between a and b; i.e., the root of f satisfies.