State the Fundamental Theorem of Algebra for polynomials with real coefficients I am working through a set of problems given to me and I have the following two questions presented to me:


*

*State the Fundamental Theorem of Algebra for polynomials with complex coefficients.

*State and prove the Fundamental Theorem of Algebra for polynomials
with real coefficients from the Fundamental Theorem of Algebra for
polynomials with complex coefficients.


I started to search the internet for both statements and when I search the second statement, I get in return the same result as the search for the first statement.
The search has provided me with this $\dots$
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with zero imaginary part.
So what is the different between both of these statements?
 A: *

*(1) means that each non constant polynomial with complex coefficients factors as product of degree $1$ (complex) polynomials

*(2) means that each non constant polynomial with real coefficients factors as product of polynomials of degree $1$ or of degree $2$ with strictly negative discriminant.

*(1) implies (2) because it shows a fortiori that any non-constant real polynomial has a complex root $z$, and by conjugation (as the polynomial has real coefficients), the conjugate $\overline{z}$ of $z$ is also a root. So complex and not real roots $z$ go by pair with conjugation, forming a degree $2$ polynomial $(X-z)(X-\overline{z}) = X^2 - 2\mathfrak{Re}(z) + |z|^2$ with discriminant $\delta = 4 \mathfrak{Re}(z)^2 - 4 |z|^2 = 4 \mathfrak{Re}(z)^2 - 4 \left( \mathfrak{Re}(z)^2 + \mathfrak{Im}(z)^2 \right) = - 4 \mathfrak{Im}(z)^2 < 0$ because $z$ is not real, leaving only real roots, and showing (2). This also shows that the irreducible polynomials with real coefficients are polynomials of degree $1$ and polynomial of degree $2$ with strictly negative discriminant.

