[A post from old and now defunct blog]
I was at a conference, and my lecture (to Foundation Studies students, that is, zero (preparatory) level) was given by my colleague Hovik, an inspirational and highly popular — although idiosyncratic — lecturer. An e-mail he sent me after the lecture was a nicest gift one can get:
Grisha helped me to find the lecture room, and I gave this lecture comparing prime numbers with the elements in the Periodic Table. Then I explained that there are only a hundred elements, while there are much more prime numbers than that.
I still managed to conclude the lecture by proving that, unlike the elements, there are infinitely many prime numbers.
Meanwhile, 2000 miles away from Manchester, on the same day and the same time, I had a chat with another my colleague, Oleg. He told me about a conversation he had with some graduate students earlier that day. Oleg was telling them about an old open problem in ring theory:
Can an infinite finitely presented associative ring be a skew field?
One of the students asked: “and what about the commutative case?” “Of course, an infinite field cannot be finitely generated as a ring” – answered Oleg. “Why?” — continued to insist the students.
And this was a typical situation when a mathematician had to resort to a specific tool in his arsenal, tool very much neglected and almost never discussed in the literature: quick proof in a special case. Indeed, the students did not expect a full proof; they would be quite satisfied with a quick demonstration in some special case. The trouble is, when you encounter such a problem, you usually have to produce a solution on the hoof.
And Oleg found one. Let us look at the simplest case: the field Q of rational numbers. Why cannot it be generated, as a ring, by finite number of numbers, say, r1, …, rn. In plain language — why cannot be every rational number expressed as a polynomial in r1, …, rn, that is, by means of addition and multiplication only, with no division? An explanation is very simple. Write every ri as a fraction — ratio of two integers — ri = pi/qi. There are only finitely many of prime numbers appearing in the denominators q1, …, qn. When you add or multiply fractions, you cannot get new prime factors in the denominator — they are those of the summands or multiplicands. Hence the conclusion:
The field Q of rational numbes cannot be generated as a ring by finite number of fractions because there are infinitely many prime numbers.
A wonderful coincidence on a lucky day — the two stories had the same mathematical content.