Rationals and the Square Root of 17

Lets think about a simple example. What is the smallest field that contains the rationals and $\sqrt{17}$? We write it as $\mathbb Q(\sqrt{17})$. In general, we would write $k(\alpha)$ to mean the smallest field containing $k$ and $\alpha$. So what is $\mathbb Q(\sqrt{17})$? Clearly, it must have every $a+b\sqrt{17}$, for all rational $a$, $b$. It turns out that this is exactly it:

$\{a+b\sqrt{17}\mid a,b\in\mathbb Q\}=\mathbb Q(\sqrt{17})$

Let’s check. We just need to make sure that we can invert any element other than zero. So if we have some $a+b\sqrt{17}$,

$\frac1{a+b\sqrt{17}}=\frac{a-b\sqrt{17}}{(a+b\sqrt{17})(a-b\sqrt{17})}=\frac a{a^2-{17}b^2}-\frac b{a^2-{17}b^2}\sqrt{17}$

This is of the form $x+y\sqrt{17}$, so we are done! While I don’t want to check it here, it is in general true that if $\alpha$ is a root of a polynomial $p(x)$ with coefficients in a field $k$, then $k(\alpha)=\{c_0+c_1\alpha+c_2\alpha^2+\cdots+c_n\alpha^n\mid c_i\in k\}$ for some big enough $n$.

For a moment let’s forget about how to multiply in general in $\mathbb Q(\sqrt{17})$. 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! $\mathbb Q(\sqrt{17})$ is a vector space over $\mathbb Q$! So what is its dimension?

If you guessed 2, you guessed correctly. Clearly $1$ and $\sqrt{17}$ are linearly independent over $\mathbb Q$ (that is, $\sqrt{17}$ is irrational), so its at least dimension 2. Furthermore, $\{1,\sqrt{17}\}$ span all of $\mathbb Q(\sqrt{17})$, so the dimension is at most 2, making it exactly 2.

You may have noticed that there was nothing special about $\sqrt{17}$ here. We could have picked $\sqrt r$ for any rational $r$. Well, almost any rational. It would be dumb to pick 9, since $\sqrt{9}=3\in\mathbb Q$. But so long as the square root is not rational, $\dim_{\mathbb Q}\mathbb Q(\sqrt r)=2$.

Mathematicians will sometimes write this as $[\mathbb Q(\sqrt{17}):\mathbb Q]=2$. Why? It makes tomorrow’s result look slightly nicer, and kind of mirrors other notation in group theory. Quit asking questions!

3 Responses to Rationals and the Square Root of 17

1. Pingback: The Tower Rule « Andy Soffer

2. JCummings says:

Typo: Do you mean $\{a + b\sqrt{17} | ... \}$ as your first set and $a + b\sqrt{17}$ at the end of your next sentence?

• soffer801 says:

Why yes i did. i originally had $\sqrt2$ and ended up changing everything to $\sqrt{17}$ to make it clear that the 2-dimensionality was not from the two, but the square root. I missed a few, and made some other mistakes. Thanks Jay.

PS, post more often. I want more Ramsey Theory!