Kronecker's theorem

We apply the above results to proving that every polynomial has a root in some (extension) field.

Theorem 2.6.11 (Kronecker's theorem)   Let $F$ be a field and let $p(x)\in F[x]$ be a non-constant polynomial. There is a field extension $E$ of $F$ and an element $c\in E$ such that $p(c)=0$ (in $E$).

proof: We may assume that $p(x)$ does not have a linear factor in $F[x]$ since otherwise that factor would yield the desired root (in $F$, which is an extension field of itself!). Let $f(x)$ denote a factor of $p(x)$ which is irreducible. Let $E=F[x]/(f(x))$. This is an extension field of $F$. The representative $x\in F[x]$ of the element $\overline{x}\in E$ satisfies $p(x)=f(x)g(x)\equiv 0\ ({\rm mod}\ f(x))$, so $p(\overline{x})=u\cdot f(\overline{x})= \overline{0}$. $\Box$

Example 2.6.12   If $F=\mathbb{R}$ and $p(x)=x^2+1$ then $p(x)$ has a root $i$ in $\mathbb{C}\cong \mathbb{R}[x]/(x^2+1)$ (which we see upon identifying $i$ with $x$).

Exercise 2.6.13   Construct an extension field of $\mathbb{F}_2$ in which $x^2+x+1\in \mathbb{F}_2[x]$ has a zero.

Exercise 2.6.14   Show that the smallest extension field of $\mathbb{F}_2$ in which $p_1(x)=x^4+x+1\in \mathbb{F}_2[x]$ has a zero is isomorphic to the smallest extension field of $\mathbb{F}_2$ in which $p_2(x)=x^4+x^3+1\in \mathbb{F}_2[x]$ has a zero. (Hint: $p_1(x)$ is the ``reciprocal'' of $p_2(x)$: $x^4p_1(x^{-1})=p_2(x)$.)

Exercise 2.6.15   In Example 2.6.7, show that $(a+bx)^{-1}={a-bx\over a^2+b^2}$.

Exercise 2.6.16   In Example 2.6.8, show that $(a+bx)^{-1}={a-bx\over a^2+b^2}$.

Exercise 2.6.17   Let $m(x)=x^2+x+1\in \mathbb{F}_2[x]$. Let $F=\mathbb{F}_2[x]/(m(x))$. Write down the addition and multiplication tables for $F$. Show that $F$ is a field.

Exercise 2.6.18   Let $m(x)=x^2+x+1\in \mathbb{F}_2[x]$. Let $F=\mathbb{F}_2[x]/(m(x))$. Show that $F$ is isomorphic to the field in Example 2.1.19.

Exercise 2.6.19   State and prove a variant of Euler's Theorem 1.8.4. for polynomials in $F[x]/(m(x))$, where $m$ is in $F[x]$ and $F$ is a finite field. (Hint: What's the analog for Euler's $\phi$ function for polynomials?).

Exercise 2.6.20   Let $p_1(x)=x^3+x+1$ and $p_2(x)=x^3+x+2$ in $\mathbb{F}_5 [x]$. Compute $(x+2)^{2114}$ in $\mathbb{F}_5 [x]/(p_1(x))$ and in $\mathbb{F}_5 [x]/(p_2(x))$. [Hint: one of $p_1$ and $p_2$ is irreducible, the other isn't. Use Exercise 2.6.19.]

Exercise 2.6.21   In high school algebra you learned how to ``rationalize the denominator"; for instance, $\frac{1}{2+\sqrt{3}}$ is $\frac{2-\sqrt{3}}{4-3}=2-\sqrt{3}$. Express $\frac{1}{2\cdot 4^{1/3}-3}$ in terms of $2^{1/3}$. [Hint: What does this have to do with the inverse of $2x^2-3$ in $\mathbb{Q}[x]/(x^3-2)$? (and you know how to find this quickly, right?)]