305 -7. Extension fields revisited

April 3, 2009

1. Greatest common divisors.

Let’s conclude the discussion from last lecture.

If ${{\mathbb F}}$ is a field and ${p(x),q(x)\in{\mathbb F}[x]}$ are nonzero, then we can find polynomials ${\alpha(x),\beta(x)\in{\mathbb F}[x]}$ such that ${\alpha p+\beta q}$ is a gcd of ${p}$ and ${q.}$

To see this, consider ${{\mathcal A}=\{{\rm deg}(a(x)):0\ne a(x)\in{\mathbb F}[x]}$ and for some polynomials ${\alpha,\beta\in{\mathbb F}[x],}$ we have ${a=\alpha p+\beta q\}.}$

We see that ${{\mathcal A}\ne\emptyset,}$ because both ${p}$ and ${q}$ are nonzero linear combinations of ${p}$ and ${q,}$ so their degrees are in ${{\mathcal A}.}$ Each element of ${{\mathcal A}}$ is a natural number because ${{\rm deg}(a)=-\infty}$ only for ${a=0.}$ By the well-ordering principle, there is a least element of ${{\mathcal A}.}$

Let ${n}$ be this least degree, and let ${g=\alpha p+\beta q}$ have degree ${n.}$

First, if ${s\in{\mathbb F}[x]}$ and ${s\mid p,q}$ then ${s\mid \alpha p+\beta q,}$ so ${s\mid g.}$

Second, by the division algorithm, we can write ${p=gm+r}$ for some polynomials ${m,r\in{\mathbb F}[x]}$ with ${{\rm deg}(r)<{\rm deg}(g).}$ Then ${r=p-gm=(1-\alpha m)p+(-\beta m)q}$ is a linear combination of ${p,q.}$ Since ${{\rm deg}(r)<{\rm deg}(g),}$ and ${n={\rm deg}(g)}$ is the smallest number in ${{\mathcal A},}$ it follows that ${{\rm deg}(r)=-\infty,}$ i.e., ${r=0.}$ This is to say that ${p=gm,}$ so ${g\mid p.}$ Similarly, ${g\mid q.}$

It follows that ${g}$ is a greatest common divisor of ${p,q.}$

Since any other greatest common divisor of ${p,q}$ is ${ig}$ for some unit ${i,}$ it follows that any gcd of ${p}$ and ${q}$ is a linear combination of ${p}$ and ${q.}$

Notice that this argument is very similar to the proof of the same result for ${{\mathbb Z}.}$

March 5, 2009

Last lecture we defined ${{\mathbb F}^{p(x)}}$ where ${{\mathbb F}}$ is a subfield of a field ${{\mathbb K},}$ all the roots of the polynomial ${p(x)}$ are in ${{\mathbb K},}$ and all the coefficients of ${p(x)}$ are in ${{\mathbb F}.}$ Namely, if ${r_1,\dots,r_n}$ are the roots of ${p,}$ then ${{\mathbb F}^{p(x)}={\mathbb F}(r_1,\dots,r_n),}$ the field generated by ${r_1,\dots,r_n}$ over ${{\mathbb F}.}$
The typical examples we will consider are those where ${{\mathbb F}={\mathbb Q},}$ ${{\mathbb K}={\mathbb C},}$ and the coefficients of ${p(x)}$ are rational or, in fact, integers.