From Tumbolia:

You can write a regular expression that matches only strings of a composite number of x’s. (Hint: Too easy for a hint.)

Ok, this looks easy enough – but dang it, it’s 11.30, and I need to go sweep the house, swab down the floors, and do my bit to dispel the general air of run-downness that’s been hanging over the dwelling for the past I don’t know how many weeks. Will be back, tomorrow.

And no, I can’t think of a regexp that does that – yet.

QOTD

The last words of Evariste Galois – Ne pleure pas, Alfred! J’ai besoin de tout mon courage pour mourir à vingt ans!

Translation – Don’t cry, Alfred! I need all my courage to die at twenty!

Stuck.

I’m stuck. This is probably trivial, but I’m just not able to get it. Yet.

Consider , with addition and multiplication defined thus – , the quantities in parentheses being ‘ordinary’ addition and multiplication. Halmos [1] then asks his readers to show that is a field if and only if is prime.

It’s fairly straightforward to show that if is a field, must be prime – one way to do this is to show that if is composite, then any that divides will not have a multiplicative inverse. Another way is even simpler – is the minimum number of times must be added to itself to produce . Consider . Now, . Also, . This means that either or , which is a contradiction. (The second “proof” is more elegant than the first, but is unfortunately not mine.)

I’m stuck trying to show that being prime implies that is a field. The only thing that needs to be done is to show that a multiplicative inverse exists for all ; in other words, such that .

Let’s see, there’s only a day’s work left before the weekend. Maybe a strong shot of coffee on a chilly Saturday morning will do the trick. I peeked into Halmos for a short respite from a few dry sections in H&K, and look at what happened. It does feel a little embarrassing to admit that I’ve spent more than a day on this. Sigh. Okay, not quite a day – because the only time I get to spend thinking about this stuff is early in the morning and late in the evening. Ha. Sigh. Oh, the things that I left undone when I was young.

Update [06 Jan 2008]:

Nothing yet. Thought about this half-heartedly for some time. Trying to resist the temptation to google for the answer. Crap, it looks like you either learn these things when you’re studying, or you don’t at all. Let’s see. Anyway, for the time being I’m just going to go ahead. I’m happy to know that the characteristic of a field is either zero or a prime number.

Update [09 Jul 2008]:

Thanks for your comment, Guppy! Both the times that I tried to respond in a comment, the comment got mangled. Here’s what I was trying to write:

I was thinking of a proof using Euclid’s algorithm, like so:

, prime and .

Hence we can find integers such that

The rest follows… but I’m not really happy with this proof, I think I’ll give the Guppy Method a go.

—
[1] Paul R. Halmos. Finite Dimensional Vector Spaces. Springer, 1974.

Closure! Closure! Closure – Argh :-/

Argh. Much against the ministrations of 1-cyclopropyl-6-fluoro-4-oxo-7-piperazin-1-yl-quinoline-3-carboxylic acid, N-(4-Nitro-2-phenoxyphenyl)methanesulfonamide and the other things that the Doctor prescribed, I’m not asleep yet. Well, in a little while I will be – but I need to get this rant out of my system before I join company with Morpheus. I would also like to rant about the brainless bastards in the neighbourhood who believe in celebrating religious (and other) occasions by unleashing a high-decibel Himesh Reshammiya assault that drowns out the sound of everything else. I’ll save that for later, however.

Warning: I am mathematically illiterate, more or less.

I was idly reading through my ancient copy of Hoffman & Kunze (finally!), and exercising my ageing grey cells with some of the exercises, when something got me worried. There was one exercise (a trivial one, going by what I’d scribbled in the margin ages ago) that required the reader to prove that every field of characteristic zero contained a copy of the rational number field. Now, this is trivial to prove if the definition of a field includes the property of closure – namely, . (It does, by the way.). The trouble was, I’d neglected to read through the first page-and-a-half in Chapter 1 of H&K properly, and was left thinking that a field need not necessarily have closure. I wasted a day in thinking of ways to prove the damn thing without assuming closure, before I came to the conclusion that it was impossible. I then did what I should have done two days ago – sieved through the text with a fine tooth-comb to discover that H&K indeed mention closure, but not in so few words.

This is the kind of stupid mistake that would make a professional mathematician throw up upon hearing of it. Right now, I feel like hitting myself on the head with a stick. Or maybe a heavy large-print hardbound copy of H&K. One of the greatest regrets that I have is not taking up RVR’s linear algebra course – in hindsight, perhaps it was just as well – my being a student in his class would’ve probably given the poor man a coronary.

Links for Today

Some idle surfing today yielded these two awesome links. Gleaned both of them off God Plays Dice, which is itself an awesome blog :) :

Walter Lewin’s Physics Lectures

The Secret Blogging Seminar.

Wow.