3 – Kronecker’s theorem

(This is part of my series Field theory for high-schoolers.)

This post states a fundamental result about polynomials, due to Kronecker, which is well-known in the case of the extension $\mathbf{C}/\mathbf{R}$, but is also valid in a more general setting. It really belongs in post 2, but since post 2 is quite long and because it is such an important result, I decided to put it in a separate post.

We have seen that it is possible for a polynomial with coefficients in a field $F$ to have no root in $F$, but have roots in an extension of $F$. The classical example is $x^2+1$ in $\mathbf{R}$: complex numbers were invented precisely to “give it” a root.

Actually, every non-constant polynomial in real coefficients has a root in $\mathbf{C}$ (since a polynomial in real coefficients can also be viewed as a polynomial in complex coefficients, and $\mathbf{C}$ is algebraically closed). This is an illustration of a more general result due to Kronecker, which states that given a field $F$ and a non-constant polynomial $f(x)$ with coefficients in $F$; there always exists an extension $E$ of $F$ such that $f(x)$ has a root in $E$.

Of course, if $f(x)$ has a root in $F$, we are done because $F$ is already the desired extension. The result is more impressive (and useful) in the case where $f(x)$ has no root in $F$. In the rest of our work, we will describe the nature of $E$.

Next: Algebraic elements

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