1 – Roots of a polynomial

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

The reader is probably familiar with the process of finding roots of a polynomial with real or complex coefficients. This post can seem to merely be an exercise in describing simple things in a complicated way, but remember that we are only applying those ideas to the simple case $\mathbf{C}/\mathbf{R}$. In a more general setting, the situation is more complicated, and understanding of the notions introduced here is crucial.

1.1 Roots of a polynomial

As described in the previous post, given a polynomial $f(x)$ with coefficients in a field $F$, we can evaluate it at any element $\alpha$ of any extension $E$ of $F$, yielding an element $f(\alpha)$ of $E$. Of particular interest are the elements $\alpha$ of $E$ such that $f(\alpha) = 0$. Those elements are called the roots (or zeroes) of $f(x)$ in $E$.

Examples:

  • The polynomial $f(x) = x^2-4$ with coefficients in $\mathbf{R}$ has two roots in $\mathbf{R}$ (namely, $2$ and $-2$).
  • The polynomial $g(x) = x^2-ix+1$ with coefficients in $\mathbf{C}$ has two roots in $\mathbf{C}$.
  • The polynomial $h(x) = x^2+1$ with coefficients in $\mathbf{R}$ has no root in $\mathbf{R}$, but has two roots in $\mathbf{C}$.

This last example shows that it is possible for a polynomial with coefficients in some field $F$ to have no roots in $F$, but have roots in some extension of $F$. This fact will be crucial in the work that follows.

1.2 Algebraically closed fields

A field $F$ such that every non-constant polynomial with coefficients in $F$ has a root in $F$ is said to be algebraically closed. The last example above shows that $\mathbf{R}$ is not algebraically closed. It might be known to the reader (but it is not easy to prove!) that $\mathbf{C}$ is algebraically closed.

Next: Irreducible polynomials

Questions:

  • Find a root of $x^2+1$ in $\mathbf{F}_2$.
  • Find a root of $x^3+x+1$ in $\mathbf{F}_3$.
  • Find two roots of $x^2+1$ in $\mathbf{F}_5$.
  • Is $\mathbf{F}_5$ algebraically closed?
  • In general, is it possible for a field with only a finite number of elements to be algebraically closed?

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