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}.}

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305 -Extensions by radicals (2)

March 5, 2009

2. Extensions by radicals

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.

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305 -5. Extensions by radicals

March 4, 2009

(This post was typeset using Luca Trevisan‘s LaTeX2WP program.)

Last lecture we characterized subfields and used the characterization to provide many new examples of fields. Now we start to explore systematically which subfields of the complex numbers are suitable to study the question of which polynomial equations can be solved. Read the rest of this entry »