MA20217 Algebra 2B (Spring ’12)

Tutorials

I give tutorials for Group 1 at 14:15 on Mondays in 8W 2.15, and for Group 9 at 14:15 on Fridays in 1WN 3.23. Work should be handed in to my folders in the pigeonholes for the module on the first floor of 4W. Group 1 should hand work in by 1pm on the Thursday nine days after the sheets are given out, and Group 9 should hand work in by 1pm on the Tuesday one week after the sheets are given out.

Because of the awkward timetabling of the tutorials, if you’re from Group 1 then in the diary below, Week n refers to the week in which we discussed the \(n\)-th problem sheet, and the tutorial actually occurred in Week \(n+1\).

The revision tutorial will be on Friday 11^th May, at 14:15 in 1WN 3.23. It would be helpful if you could tell me what you’d like me to cover beforehand.

Course Website

The main course webpage for this module can be found here.

Week 1

On Friday we talked quite a lot about multiplicative inverses. Note that in a ring, as there is no reason why multiplicative inverses should exist for general elements, the statement “\(a^{-1}=b\)” should be interpreted as meaning “\(a\) has a multiplicative inverse, and it’s \(b\)”. So to prove such a statement, you should show that \(ab=1=ba\). You needn’t have a systematic method for computing \(b\) from \(a\), but often there will be one (as in the case of matrices).

On Monday we talked more about the mechanics of proving that things are rings. Normally a good plan is to reduce the problem to a case you know well, as in the first exercise, where you can translate a lot of the algebra from End(V) into V (which you know a lot more about) by evaluating the maps on vectors. One of you astutely observed that a similar idea saves you a lot of messing about with matrix algebra in question 2.

Week 2

On Friday we talked about calculating things in \(\mathbb{Z}_{11}\), and discovered an interesting fact related to the well-known result that if the sum of the digits of a decimal number is a multiple of \(3\), then so is the original number. You might want to try proving the following general form of this result:

“If the sum of the digits of an integer \(x\), written in base \(n\), is divisible by \(k\), then \(x\) is divisible by the greatest common divisor of \(n-1\) and \(k\). If the alternating sum of these digits is divisible by \(k\), then \(x\) is divisible by the greatest common divisor of \(n+1\) and \(k\).”

Week 3

It is important to remember that to show that the inverse of an isomorphism is itself an isomorphism, you must show that this inverse is a homomorphism (you get the ‘bijective’ part for free). This way of thinking is particularly important in some general contexts when being a bijective homomorphism is not sufficient to be an isomorphism, and definition becomes ‘a homomorphism is an isomorphism if it is bijective, and its inverse is also a homomorphism’.

Week 4

When proving that \(G=\mathbb{Z}+\mathbb{Z}\mathrm{i}\) is a principal ideal domain, you are given the hint that for an ideal \(I\) of this ring, you should let \(a\) be an element of \(I\) with \(\lvert a\rvert^2\) minimal, and show that \(I=Ga\). To show that every element of \(I\) is in \(Ga\), you should consider a general element \(b\) of \(I\), and consider the nearest element of \(Ga\) to \(b\). By using the minimality of \(\lvert a\rvert^2\), you should be able to show that this nearest element is actually \(b\), and so \(b\) is in \(Ga\).

Week 5

The following are useful things to remember for this week:

Polynomials in \(K[x]\) of degree at least four can have no roots in \(K\), but still be reducible.
In a principal ideal domain, any irreducible element is prime.
If you want to prove that a ring is a PID, see if you can think of a sensible notion of the ‘size’ of an element. If you can, then often ideals will be generated by an element of minimal size.

Week 6

One of the main lessons of this week’s exercise sheet is that it is often easier to prove statements about divisibility by working in modulo arithmetic. In particular for question 5, the argument involves working in the Gaussian integers, the regular integers, and \(\mathbb{Z}_p\) at various points—it is useful to be able to switch comfortably between these various rings. Note also how much we can get out of the result that in a principal ideal domain (or indeed even in a unique factorisation domain) prime and irreducible elements coincide.

Week 7

One of the main lessons of this week’s sheet is that you can save yourself a lot of pencil mileage when proving identities in which both sides are bilinear in their two variables by checking the identity on a basis. This is very similar (in fact, really the same) as the result that two linear maps are equal if they agree on a basis.

You have also been dealing with field extensions, and the results you have seen are some of the first results in Galois theory. Because the Galois theory unit only runs every other year at Bath (at the time of writing), if you want to find out more about this subject you have to take the course next year, which means also taking Algebraic Structures in the first semester. The main theorem tells you that if you have a field \(K\) inside a larger field \(L\) then you can write down (in a very specific way) a Galois group, normally written \(\operatorname{Gal}(L,K)\), such that under nice conditions the subgroups of \(\operatorname{Gal}(L,K)\) are in bijection with fields \(M\) which contain \(K\) but are contained in \(L\) (and in fact this bijection tells you how to translate between subgroups and these intermediate fields). This type of correspondence is known as a Galois correspondence and turns up a lot whenever you are thinking about ‘extending’ objects. One example of this, which you can understand when you do some topology, is that the path-connected covering spaces of a topological space \(X\) correspond to the subgroups of the fundamental group of \(X\), which here plays the role of the Galois group.

Week 8

The final exercise on the eighth example sheet reprises the final exercise on the sixth, and continues the investigation into which elements in the ring of Gaussian integers are prime. You may find it useful to look back at what happened in the previous instalment.

Weeks 9–10

I won’t be here during these weeks, so your tutorials will be taken by Elvijs Sarkans, except for the tutorial for Group 1 on Monday 17^th April, which will be taken by Alex Collins.

Week 11

Make sure you know how to do Jordan normal form calculations like the one on the last sheet, as they tend to make very good exam questions. The \(4\times 4\) case should be sufficient, any bigger than that and things can start to get more complicated to work out.