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 $\alpha _{j}=a_{j}+ib_{j}$ can be paired with its conjugate ${\overline {\alpha }}_{j}=a_{j}-ib_{j}$ (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 $(x-\alpha _{j})(x-{\overline {\alpha }}_{j})=(x-a_{j})^{2}+b_{j}^{2}$, $b_{j}\neq 0$.

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 $a<\alpha _{j}<b$.

References

• R. G. Swan, Tarski's Principle and the Elimination of Quantifiers at Richard G. Swan
• Srivastava, Shashi Mohan (2013). A Course on Mathematical Logic.