## Monthly Archives: May 2008

### A few past exam questions

A few past exam questions for 3704, in response to a query by Hollie and Kayan.

### Some more classgroups

Hi Sir,

I have been working through the past papers and was hoping you could verify my results for computing class groups:

I get

$Q(\sqrt{-30})$ to be trivial;

$Q(\sqrt{-21})$ and $Q(\sqrt{35})$ to be isomorphic to $Z/2 \times Z/2$

Also are there any other numbers useful to work out the class groups for.

Many thanks,

(ps I saw the Algebra 3 exam and I thought it was a fair exam so cannot understand why so many people are complaining)

$Q(\sqrt{-30}): Z/2\times Z/2$

$Q(\sqrt{-21}): Z/2\times Z/2$

$Q(\sqrt{35}): Z/2$

Well, make sure you understand the examples on the coursework and the sample sheet. You can also try computing with any old number you want for practice. To check your answers, I urge everyone to download the program available at

The Pari/GP website

and try to figure out how to use it to compute class groups. You’ll find the command for computing just the class number, i.e., the order of the class group, in the Tutorial. But for the actual group structure, you have to look into the User’s Guide. After downloading, if you need some help, let me know.

Of course this program will not help directly with the exam, since there, you have know *how* it’s computed. But you can use it now for checking your answers.

### Generators for maximal ideals

Hello Professor Kim,

I’ve found myself stuck on finding maximal ideals when finding class groups. for example:

$m= X^3 - 2$

How do we make our decision to use

$X^3 - 2 = (X - 3)(X^2 + 3X - 1) \mod 5$

to find $P_5$ and $P_{25}$

How do we know not to use

$X^3 - 2 = (X - 3)(X^2 + 3X + 4) \mod 5$

or any other variation,e.g.,

$X^3 - 2 = (X - 3)(X^2 + 3X + 9) \mod 5$

Thanks

This is a good question! Seeing it, I can see you’re reviewing seriously. Note first of all that all the degree two polynomials you’ve listed are the same in $F_5[x]$. So the question is, which lift to $Z[x]$ does one use to write down generators for the maximal ideals? And the answer is:

Any!

In your example, the point is that

$(5, \alpha^2 + 3\alpha - 1)=(5, \alpha^2 + 3\alpha + 4)=(5, \alpha^2 + 3\alpha + 9)$

in $O_K=Z[\alpha]$, where $\alpha=2^{1/3}$. The elements are different, but the ideals listed are the same, because 5 is already included in each of them. For example,

$\alpha^2 + 3\alpha + 4=(\alpha^2 + 3\alpha -1)+5$

and hence,

$\alpha^2 + 3\alpha + 4\in (5, \alpha^2 + 3\alpha - 1).$

The maximal ideals occurring in Dedekind’s theorem all look like

$(p, f(\alpha))$

so that

$(p, f(\alpha))=(p,g(\alpha))$

whenever $f(x)\equiv g(x) \mod p$.

### One more remark

I recommend that people take the hard advice in the previous post seriously. But for what it’s worth, I will add one remark related to the “format” that seems to have perturbed at least some people. As far as I can judge, the “shape” of the 3704 exam does seem to be closer to that of previous years than the 2201 exam was. I am deliberately describing the correlation in vague terms, as you might guess. (On the other hand, my sister, who works on exactly such issues in computational linguistics, may be able to make it precise. Her concern, to a certain extent, is with the topology of documents.)

### 3704 exam

I’ve received recently several queries about the correlation of this year’s exam with those of previous years. I think I’ve discussed already the rather complex issues surrounding this query, and hence, do not believe further attempt at prediction will be helpful. On the other hand, I would strongly recommend against preparing just by passively going through the past exams.

If you’re somewhat following the discussion about 2201, you’ve probably gathered that the differences have been exaggerated. Furthermore, even though I’ve expressed my sympathy many times about the travails of the students, that sympathy hasn’t been primarily about such differences, which you could interpret as big or small depending upon your grasp of the material. Some students have perhaps done themselves a disservice by reading prior advice out of context and just in comforting portions. For 3704 as well, one email reads:

“It is stated in your Practical Summary that our exam will be the same pattern of previous years.[sic]“

The actual sentence in the practical summary says:

“Even though the material emphasized in the exam will follow the pattern of previous exams, make sure you understand examples that illustrate the general theorems.”

If you look at the 2201 examination, even the most critical reading would not conclude that the “material emphasized” was significantly different from previous years. Another possibility for a convenient misreading might be to draw an inference: “Therefore, you don’t need to know proofs. Just learn computation.” Nothing of the sort is being stated.

As I’ve said in relation to the 2201 exam, a good deal of my advice evolved from a need to counter a tendency to *memorize* proofs without absorbing the material properly. In fact, the method of learning proofs that I was actually recommending is to grasp what is “really going on” so thoroughly as to render it pointless to memorize.

If there’s one assurance I can give in relation to the past exams, it’s the same one I gave in class and, I think, in several communications with students: If you have a thorough understanding of the material necessary to solve the past problems efficiently and with confidence, you won’t have trouble this year.

### Class group, norms,…

Hi Sir.

I was looking at class groups (as one does when he/she finds the rest of the world trivial and unjust -is it possible for it to be both?)

I noticed that although we had done many examples of the form $Q(\alpha)$ where $\alpha$ was perhaps the root or cube of an integer, we hadn’t done one where $\alpha$ was the root of a (dare i say) ‘longer’ polynomial.

From my understanding, the only problem we might incur is that the ring of integers is not straightforward & that ‘s’ may also not be obvious.

Is this the only difference?

And lastly, rather a silly question. When calculating the trace of, say, $\alpha^2$ in a cubic field, is it so that the other conjugates are simply $(\zeta_3 \alpha)^2$ and $(\zeta_3^2\alpha)^2$

$=> tr(\alpha^2)= (1+\zeta_3^2+\zeta_3^4)(\alpha^2)=0??$

(Here, $\zeta_3$ is a primitive cube root of 1.)

It appears one forgets the simple things in life when constantly faced with more pressing issues.

Thanks.

I’m glad you find solace in computing class groups! Mathematics actually has great therapeutic potential.

It’s true that we’ve concentrated mostly on quadratic and cubic fields. Computing $s$ is rather easy in general. It just requires a bit of calculus to figure out the number of real roots for any polynomial of reasonable degree, and hence, the number of complex conjugate pairs of roots. Meanwhile notice that we haven’t discussed prime decompositions for any field $K$ where $O_K$ is not of the form $Z[\alpha]$, even for cubic fields. This is one difficulty. Even assuming that form, there are many other computational complexities for fields of higher degree, although degree 4 extensions of $Q$ with Galois group $Z/2\times Z/2$ are more or less tractable. You can look at Henri Cohen’s book on computational number theory to get a sense of what the challenges are. Or simply do a google search on `computing class groups,’ which should take you to some interesting references.

In the norm problem you mention, one fact that’s true in general is that if $f(\alpha)$ is any polynomial of $\alpha$ with rational coefficients, then $\sigma f(\alpha)=f(\sigma(\alpha))$ for any embedding $\sigma$. This gives the formula for the norm you wrote down *if* the minimal polynomial of $\alpha$ is of the form $x^3-a$, that is, $\alpha$ is a cube root of something. However, if the minimal polynomial of $\alpha$ is more complicated, then the conjugates of $\alpha$ are themselves more complicated, so you can’t have such a simple formula for the trace and norm of $\alpha^2$ either. However, when the minimal polynomial of $f(\alpha)$ has the form $x^3+ax+b$, i.e., no $x^2$ term, you still get $Tr(f(\alpha))=0$ for trivial reasons.

### Algebraic integers

Hi

In regards to Algebraic number theory, a couple of the past papers have the question: Show that Z[a] is the ring of algebraic integers in Q[a], a standing for alpha. I am not sure how to do this, can you advise me?

Many Thanks

Look at some of the problems in the optional coursework sheet 4.

### Integral bases for quadratic fields

Hi,

I was wondering if you could just clarify how you determine $O_K.$ In most quadratic examples it seems to be $Z(\sqrt{n})$, however in some it is $Z((1+\sqrt{n}/2))$. I think it must have something to do with the integral basis?

———————————————————————————–

I refer you to lecture 11 in the online notes. You should review this material thoroughly.

### Erratum on Jordan normal form

Several people pointed out to me an elementary misprint in
an earlier post . It’s corrected now, but it originally claimed that the Jordan normal form was determined by the characteristic
polynomial and the minimal polynomial up to $6 \times 6$ matrices. Actually, this is not true even for $4 \times 4$ matrices. (Consider, matrices $A$ with $ch_A(x)=(x-1)^4,\ \ m_A(x)=(x-1)^2$.) What I meant to say is that the JNF is determined by those two polynomials *and* the dimension of the eigenspaces (=number of Jordan blocks) for each eigenvalue. These three numbers might be referred to as the *elementary invariants* of a matrix.