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}.}

Read the rest of this entry »


305 -Homework set 7

April 3, 2009

This set is due April 10 at the beginning of lecture. Details of the homework policy can be found on the syllabus and here.

1. Let R be a commutative ring with identity. Let I be an ideal of R, and let i be a unit of R. Show that I=R iff i\in I. Conclude that the only ideals of a field {\mathbb F} are \{0\} and {\mathbb F}. Also conclude that {\mathbb F}[x]/I\cong\{0\} if I is the ideal of {\mathbb F}[x] generated by a constant nonzero polynomial.

2. Suppose p(x)\in{\mathbb F}[x] is a nonconstant, not irreducible polynomial, and let I=(p) be the ideal generated by p. Show that {\mathbb F}[x]/I has zero divisors.

3. Find an irreducible polynomial p(x) in {\mathbb Z}_2[x] of degree 3, and explicitly show that {\mathbb Z}_2[x]/(p) coincides with (i.e., is isomorphic to) the field of 8 elements built in a previous homework set.

4. Either build a field of 9 elements using the kinds of arguments in the previous problems; or determine a subfield of {\mathbb C} isomorphic to {\mathbb Q}[x]/(x^3+3x+3). (You may assume that x^3+3x+3 is irreducible in {\mathbb Q}[x].)