Gaussian Integers

1 Introduction

The ring of Gaussian integers is given by ℤ[i] = {a + ib∣a,b ∈ ℤ}. The units of ℤ[i] are {1,-1,i,-i}. The irreducible elements of ℤ[i] are the associates of

• i + 1,
• prime numbers p of ℤ such that p ≡ 3 mod 4 and
• a + ib where p = a² + b² ≡ 1 mod 4 is prime in ℤ.

Consider the factors of integers in ℤ[i].

Integer n	Factors with positive real part	s(n)
1	1	1
2	1, 1 + i , 1 - i , 2	5
3	1, 3	4
4	1, 1 - i, 1 + i, 2 - i, 2 + i, 4	13
5	1, 1 - 2i, 1 + 2i, 2 - i, 2 + i, 5,	12

The function s(n) is the sum of the positive real parts of the factors.

Proposition 1 Let k ≥ 1. Then s(2^k) = 5 ⋅ 2^k-1 + 3 ⋅ (2^k-1 - 1).

Proposition 2 Let k ≥ 1 and p ≡ 3 mod 4 be prime. Then s(p^k) = .

Proposition 3 Let k ≥ 0, p_i ≡ 3 mod 4 be primes and a_i ≥ 0 for i = 1, 2,…,r. Then s(2^k ∏ _i=1^rp _i^a_i) = s(2^k) ∏ _i=1^rs(p _i^a_i).

Proposition 4 Let p_i ≡ 3 mod 4, q_j ≡ 1 mod 4 be primes and a_i,b_j ≥ 0 for i = 1, 2,…,r and j = 1, 2,…,s. Then

s(∏ _i=1^rp _i^a_i ∏ _j=1^sq _j^b_j) = ∏ _i=1^rs(p _i^a_i) ⋅ s(∏ _j=1^sq _j^b_j).

Proposition 5 Suppose p ≡ 1 mod 4 such that p = σσ^* where σ = a+ib ∈ ℤ[i]. Then s(p) = 1 + p + 2(a + b).

I got tenure..

.. just kidding! I've been extremely lazy lately, curse the internet and BBC News. I need to start carrying out a lot of tasks in the next few weeks:

• Start reading my notes on general topology and completing the homework. Same goes for my class in finite simple groups.
  
• Rounding up my dissertation which seemed to be on halt for the past month.
• Review my french.
• Get my flatmate to fix linux on my laptop.
• Much, much more.

That's all I had to say really, oh wait, good luck to Kosovo!

Summary: 2007

By popular demand (1 and still counting) I have decided to write a summary of what happened in 2007 to make up for my lack of posts.

I will begin the high points of 2007 because after writing the low points I might not be in the mood to continue with this post. The first and most memorable event of the year was meeting with one of my best internet friends in London around the 2/3rds mark of the first quarter. It was only one day, but I thoroughly enjoyed myself that day, perhaps moreso than any other day of the year. We spent the day touring the city and its museums, and at times finding a way out of the state of being 'lost' :D.

For the next few months I soldiered along with my studies and achieved as much as my body and mind could afford. Through my diligence I was awarded with a prize and summer stipend to do research with one of my professors, which lasted approximately 2 months. For the majority of that time I was reading up on the required background information and trying to digest it, but for the last few weeks I had more freedom and my professor gave me various results to prove. We managed to prove some small result independently and we decided to write a preprint, but we haven't actually got around to writing it up yet. This is what my summer consisted of and I didn't really take much of a break before the next semester begun.

Once the next semester had started I immediatley got on with my dissertation but after about a month of failing to understand a paper as my background did not permit me to understand most of the concepts. So after about 6 weeks, we (my superivor and I) decided to slightly change the direction of my project and I begun to learn Algebraic Topology. As I had no background knowledge in topology whatsoever it was a difficult and slow start, in fact by the time the semester had ended (mid-December) I had only written around 15 pages. But throughout the Christmas `break' I decided not to take any break (except for the three days I was sick) and worked entirely on my dissertation. I managed to write on average 2.5 pages per day, which transformed my project from a brief introduction to an almost complete 67-page project. I am still waiting for my supervisor to read it though and any decision on what to do next will be made through his feedback.

I guess those are all the high points of 2007 and now we come to 2008. I haven't made any new years resolutions, as I make resolutions after each day and never seem to be able to stick to them. The low points of 2007 I would rather much avoid. But lets briefly summarise them to: I made some friends I wish I didn't and now being hunted down for my blasphemous ways :P.

I cannot promise (and is very unlikely) that I will continue to post here throughout 2008. I suppose the best you can hope for is two posts per year (and this is one of them). To all my reader(s) out there, have a good a year.

Exams are approaching!

To reader(s)/empty set:
I haven't made a post here for what seems like a lifetime ago. The last few months have been quite stressful due to my studies and among other things, but watching a couple of video blogs on youtube has inpsired me to write again.

My exams will begin mid May and end late May, I'm very concerned about them at the moment and will be focusing on them more than anything else for the next few weeks. After my exams, I must give an oral presentation on the history project I submitted a few months ago, which I am dreading since I suck badly at public speaking, but I'm sure I'll find a way. Then I might possibly get admitted for a studentship over the summer vacation, although I'm not even sure if my department still offers such a programme. If not, then I'm not entirely sure what I'll do for the summer.

As it is 0h38, I will not say much more for now, except that I am back (temporarily anyway) and that in the coming months I may or may not decide to add new posts, depending on my mood really :)

Au revoir!

Free!

I am free!

nous commençons!

Classes have resumed and the skin pigmentation under my eyes is starting to show it. Yesterday I arrived at school around 10h and didn't leave until 18h35 (my last class finished at 18h); I suppose it wasn't too bad as I only had a few lectures but I almost fell asleep in my last lecture (Mathematical Finance). The first day is usually very unruly, because most students have a compulsive disorder in which they must share every detail with their friends with regards to what they have done during the winter break, and they just won't shut up during lectures.

Today I have two hours of computer labs, which ought to be fun, and then I have a late lecture. I'm still slightly worried about my project despite the claim of completion I made in my previous post. I keep changing the style of referencing and now I'm considering changing the content of the last chapter, as I'm not sure whether it fairly reflects the work of Galois. I still have over two weeks to 'complete' it, but I don't want it hanging over my head.

Are we soon to see the deunification of the united kingdom? Both Northern Ireland, Scotland and Wales have either their own parliament or an assembly, and now 61% of people in England want an English Parliament. This could mean the monarchy may soon become irrelevant and Pandeus' plans would have succeeded, rejoice! This will be his greatest achievement since that time he invaded Norway after attending a nationalist sing-along.

My computer lab session appears to be cancelled but now I must study other subjects, so au revoir!

Completion, Hooray!

I will begin by issuing the following statement: British police officers do NOT lynch minorities, I said that only for comical value and I apologise for the offense I may have caused anyone.

On a better note; I have pretty much completed my history project - I am only left with writing an abstract for an oral presentation I must give in the summer. I fear that I had exceeded the word limit so I had to reduce it a little bit, but I think it's fine now. Through my research on the theory of equations, I have come to the conclusion that the question of solvability is still open. Even though we've inherited a beautiful theory from Galois which provides us with an algorithm enabling one to determine whether or not an equation is solvable by radicals. However this algorithm can, at times, be difficult because it relies on knowing the roots of the equation. He even admits that his methods are sometimes impractical. One might wonder if one can determine the solvability of an equation by simply looking at its coefficients, or perhaps this is not possible and what Galois did was the best that can be done.

Nevertheless, I have discovered that Galois Theory has a much greater importance that is not restricted to the theory of equations, from it emerged Group Theory and more generally the study of algebraic structures. If you read his work (or at least a translation) Mémoire sur les conditions de résolubilité des équations par radicaux, you will find that most of the results have no mention of equations - perhaps this was his intention. In his memoir he writes:

"Jump above calculations; group the operations, classify them according to their complexities rather than their appearance; this, I believe, is the mission of future mathematicians; this is the road on which I am embarking in this work."

I probably won't upload my project yet, as it has yet to be graded - but I will show people on an individual basis. Anyone who wishes to see it, then you may request to, but you will be required to undergo intense screening in order to determine you're not a fellow classmate nor an associate of a fellow classmate (even though we all have different projects to do).

Now that my project is essentially complete I must proceed with other work I planned to do over the winter break. Such as my combinatorics homework (which I had completed most of before the break begun) and also I intended to learn how to use Matlab before my classes in computational linear algebra begin, but now I only have about 3 days to do this. It matters not though, I already know how to use C++ for numerical problems, and so I should pick up Matlab relatively quickly.

There is still no news of my bicycle. I was struck by the sirens of a police car, speeding down the road, hoping that they had found a lead on my case, but I received no call later that day - so I figured not. I honestly don't think my bicycle will be returned now; I have come to accept this.

I will now make my way into campus to continue my studies, so until then Sayonara!

A sad day for humanity

Hello reader(s): I've had a pretty good time since my last post. In fact I managed to understand what Lagrange was trying to say, and to an extent I understand what Ruffini and Abel did; however I still must understand Galois Theory, it seems a lot to digest in only a few days - so perhaps I can get away with just learning key ideas without really reading all the lemmata/theorems. I still have this last chapter to write until I'm pretty much home and dry, except for an introduction and conclusion, I suppose this shouldn't consume too much of my precious time.

The perfect start to friday, I had. I rose from bed before the sun rose from its, and I reached my department before 10am, and proceeded with my work. I then went to leave at 12.30pm, only to discover my bicycle had been stolen. :( The culprit had cut right through my lock (a cable) and now I am bicycleless! I reported it to the security but they said it's not very likely they will be able to retrieve, and since I don't have insurance I can't claim on it and so I'm out of a lot of money. I demanded they hunt the evildoer down then amputate his hands, but they were not interested! I will report this incident to the police today, although I doubt they will achieve much; they will be too busy involved with real crimes - like handing out parking tickets or lynching minorities, damn them!

Anyway, this is why my day somewhat sucked. My faith in humanity has all but disappeared. I know this person will eventually get what's coming to them, but until then let him/her enjoy the bicycle!

Me and my equations.

Good news everyone, the cooking stove is now working! I lived off coffee, cereal, toast and take-out food for past 3-4 days. I have no idea how it was repaired, neither my flatmate or myself reported it to management; verily it was a miracle. Rejoice! I will now return to my diet of fish, rice, vegetables, coffee and bread; and occasionally Pakistani and Malaysian food. I'm also thinking about adding egg to my palette, but each time I try to fry or boil an egg it usually ends up burnt or uncooked respectively.

Another problem I must solve is understanding Lagrange's treatise "Reflexions sur la resolution algebrique des equations". His methods for explaining why cubic and quartics are solvable by equations are somewhat unclear to me. I understand each step of his work, I am just unable to link it to the problem he claimed to solve. To a degree Ruffini and Abel cleared the matter up, but I still want to comprehend what Lagrange was attempting to do. Oh I forgot to mention, this is for my history project - I'm reporting on the development of the theory of equations, namely on their solvability. I'm nearing completion, I just need to discuss the contributions of Abel and Galois, and then write introduction/conclusions. Hopefully I can get this finished before the end of the week, which will give me one more week to prepare for next semester.

I got a new (used) hat yesterday! My friend was wearing a trapping hat, so I asked him where he got it from. Then when I told him I really liked it, he gave it to me - very generous! Although I feel that it is a little tight around my head, I think I'll have to buy a bigger size to contain my painlessly. Nevertheless, it's a really cool hat and it keeps me warm during the winter mornings.

Okay, well I'm going to make myself some breakfast now - probably some toast and coffee, then I will head off to see if my department is open; if it is not then I'll go to another department with a computer cluster in order to finish project.

A Merry Old Year!

Yes, 2007 is upon us now, another chance to put things right. A chance to make new years resolutions and stick to them for at least 2 weeks, a chance to reflect on 2006 and learn from mistakes and a chance to.. do whatever your heart desires!

I spent last night working on my essay, which kind of sucks but at least I did something productive and so didn't feel guilty about wasting time (as I usually do). I was expecting to see fireworks outside, but I saw them not! I didn't even hear people in my block of flats celebrating, perhaps the spirit of new years is dying. In any case, it worked out well for me not having flatmates walking in at 3h falling all over the place and disturbing me during my sleep!

I had an enjoyable 2006, not without its stress though, however I feel I came close to fulfilling my potential for the year. It was probably my best year for quite some time. I have compiled an incomplete list of my new years resolution, in order of importance:

1. No procrastinating.
2. To pass all my exams with an average worthy of a scholarship.
3. To improve my French.
   
4. To consume at least 2 meals a day.
5. To make more of an effort to talk to people.
6. To drink less coffee.

I'll be happy if the first two are accomplished, the others are a bonus!

I hope you all have yourself a good year! I now must make some coffee, sleep and complete my essay in some undetermined random order.

P.S. I think I will make a post about my project soon; as it is far more interesting than my life :D

Home away from home.

I have departed home, and returned to my second home (student halls). It's not a bad feeling, although I'm slowly starving to death due to my ineptness in the kitchen. I Suppose there are both pros and cons living away from home.

Pros:

1. Freedom.
2. Freedom.
3. Close to my department.
4. Less likely to go into lazy mode.
5. I have to move out some day, good life experience.

Cons:

1. Must do laundry.
2. Must Cook.
3. Must share a kitchen with disgusting roommates who don't understand the concept of cleanliness.
4. It's very expensive.
   

I must snap out of lazy mood soon due to a major assignment pending completion. It carries 40% for one of my classes and it's not looking promising at this moment in time. I'm sure if I put my head down for the next week, I will manage to complete it.

In other news, we'll (the UK) have finally paid of our WWII debts to the US and Canada by the close of 2006. Final installments will be paid off today, and we'll be free again!

*...waits for the withdrawal of troops from Iraq*

Haha, well maybe not completely free! Nevertheless that's one less loan to worry about. Now all we must do is cut funding for alcohol, smoking and generally self-afflicted related diseases treatment and we're half way there to paying my school fees for the next couple of years. They should make me chancellor of the exchequer so that I can write the next budget.

Anyway I must make some coffee now, have a good day!

Happy...

...25th December! It was a grand old day. However I spent most of it in my room with a splitting headache to my great dismay! I received mostly garments to cover my body, and two other gifts which I will exchange for something more useful (through my mothers willingness). The food was rather tasty, although I didn't eat until around 19h00, we had chicken instead of turkey to my delight!

But other than that, it was a waste of a day (as predicted in point 5 here), a day in which I hardly studied at all. In fact, I have studied very little for the past 3 days I have been here. Once I return tomorrow, I must quickly re-establish my strict work ethics and continue my intense programme of pure mathematics. The history (maths) project is due in 6 weeks I believe, but I intend to finish before the next semester begins (less than 3 weeks) and looking at my current work rate, this seems highly unlikely. I have all the information avaliable to me, I've just been extremely slow (and lazy) in extracting it.

On that note, Merry Christmas to all, and to all a good morning! (it's 8h48)

Return of the Jedi

I have returned home for a few days, for CHRISTMAS! What is the meaning of christmas? According to reference.com:

"the annual festival of the Christian church commemorating the birth of Jesus: celebrated on December 25 and now generally observed as a legal holiday and an occasion for exchanging gifts."

My understanding is that the year of his birth is not even agreed upon, let alone the day! Alas this didn't divide the christian church. I miss the good old days, with the religious intolerance and persecution. Where did they go? Where are the witch hunts? I'll tell you where! They're poisoned by capitalist toxic fumes, the same fumes which engulf the christmas spirit! That's where!

I'm not a christian, but at least if this holiday was strictly a religious one, it wouldn't be as annoying as it is in present day - perhaps we could even make it a private celebration instead of a public one by claiming it's discriminates against non-christians. I will return to my home away from home in a few days, so there is light at the end of the tunnel for me.

Christmas!

Why must Christmas be celebrated every year ? Why can people not just celebrate it every 4 years like the football world cup? Don't get me wrong, the decorations are aesthetically pleasing to the eye and they remind me of good times when I was younger; however now the dreaded event becomes a nuisance. Justifications for disliking Christmas:

0. Everyone expects you to celebrate it.
1. My department closes down and is only accessible to postgraduate students.
2. Expectations of you to buy great gifts (or just gifts) for family and friends.
3. Meeting distant family.
4. Not being able to sleep early because family comes around to your house late at night.
5. You'll more than likely end up wasting the day.

I haven't decided whether I will return to my family this year to 'celebrate' it, which seems rather cold of me but I have plenty of work that needs doing before the next semester begins. Perhaps spending Christmas alone will have a profound effect on me and possibly provide me with a new appreciation for it, only time will tell.