In mathematics, the Weierstrass Nullstellensatz is a version of the intermediate value theorem over a real closed field.
It says:[1][2] Since F is real-closed, F(i) is algebraically closed, hence f(x) can be written as
α
is the leading coefficient and
α
are the roots of f. Since each nonreal root
α
can be paired with its conjugate
α ¯
(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
α
α ¯
If f changes sign between a and b, one of these factors must change sign.
is strictly positive for all x in any formally real field, hence one of the linear factors
α
α
, must change sign between a and b; i.e., the root
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