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

Roughly, the proof goes by constructing a root of a polynomial.

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.