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!

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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!

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