Rationals and the Square Root of 17
October 12, 2011 3 Comments
Lets think about a simple example. What is the smallest field that contains the rationals and ? We write it as . In general, we would write to mean the smallest field containing and . So what is ? Clearly, it must have every , for all rational , . It turns out that this is exactly it:
Let’s check. We just need to make sure that we can invert any element other than zero. So if we have some ,
This is of the form , so we are done! While I don’t want to check it here, it is in general true that if is a root of a polynomial with coefficients in a field , then for some big enough .
For a moment let’s forget about how to multiply in general in . We still know how to add. Let’s even say we still remember how to multiply by rational numbers. Well then, what do you know! is a vector space over ! So what is its dimension?
If you guessed 2, you guessed correctly. Clearly and are linearly independent over (that is, is irrational), so its at least dimension 2. Furthermore, span all of , so the dimension is at most 2, making it exactly 2.
You may have noticed that there was nothing special about here. We could have picked for any rational . Well, almost any rational. It would be dumb to pick 9, since . But so long as the square root is not rational, .
Mathematicians will sometimes write this as . Why?
It makes tomorrow’s result look slightly nicer, and kind of mirrors other notation in group theory. Quit asking questions!