In mathematics, the Weierstrass Nullstellensatz is a version of the intermediate value theorem over a real closed field. It says:
Given a polynomial f {\displaystyle f} in one variable with coefficients in a real closed field F and a < b {\displaystyle a<b} in F {\displaystyle F} , if f ( a ) < 0 < f ( b ) {\displaystyle f(a)<0<f(b)} , then there exists a c {\displaystyle c} in F {\displaystyle F} such that a < c < b {\displaystyle a<c<b} and f ( c ) = 0 {\displaystyle f(c)=0} .
Proof Since F is real-closed, F(i) is algebraically closed, hence f(x) can be written as u ∏ i ( x − α i ) {\displaystyle u\prod _{i}(x-\alpha _{i})} , where u ∈ F {\displaystyle u\in F} is the leading coefficient and α j ∈ F ( i ) {\displaystyle \alpha _{j}\in F(i)} are the roots of f. Since each nonreal root α j = a j + i b j {\displaystyle \alpha _{j}=a_{j}+ib_{j}} can be paired with its conjugate α ¯ j = a j − i b j {\displaystyle {\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 − α j ) ( x − α ¯ j ) = ( x − a j ) 2 + b j 2 {\displaystyle (x-\alpha _{j})(x-{\overline {\alpha }}_{j})=(x-a_{j})^{2}+b_{j}^{2}} , b j ≠ 0 {\displaystyle 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 {\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 x − α j {\displaystyle x-\alpha _{j}} , α j ∈ F {\displaystyle \alpha _{j}\in F} , must change sign between a and b; i.e., the root α j {\displaystyle \alpha _{j}} of f satisfies a < α j < b {\displaystyle a<\alpha _{j}<b} .
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