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

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

Proof

Since F is real-closed, F(i) is algebraically closed, hence f(x) can be written as ${\displaystyle u\prod _{i}(x-\alpha _{i})}$, where ${\displaystyle u\in F}$ is the leading coefficient and ${\displaystyle \alpha _{j}\in F(i)}$ are the roots of f. Since each nonreal root ${\displaystyle \alpha _{j}=a_{j}+ib_{j}}$ can be paired with its conjugate ${\displaystyle {\overline {\alpha }}_{j}=a_{j}-ib_{j}}$ (which is also a root of f), we see that f can be factored in F as a product of linear polynomials and polynomials of the form ${\displaystyle (x-\alpha _{j})(x-{\overline {\alpha }}_{j})=(x-a_{j})^{2}+b_{j}^{2}}$, ${\displaystyle b_{j}\neq 0}$.

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

References

1. Swan, Theorem 10.4. harvnb error: no target: CITEREFSwan (help)
2. Srivastava 2013, Proposition 5.9.11.

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

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