Rationalizing the denominator

Remember how last time I found that returning to old problems spurred on current thinking? Well it has happened again. This time returning to Alevel maths has turned up something interesting.

We learn how to rationalize the denominator of a fraction involving square roots quite early on in the Alevel course. For most this is really just a rule you apply arbitrarily to the one situation but really there is so much more to it.

Actually it can easily be seen that for any \alpha/\beta such that \beta is algebraic over \mathbb{Q} then multiplying top and bottom by the product of \sigma(\beta) as \sigma runs through all non-identity elements of Gal(\mathbb{Q}(\beta)/\mathbb{Q}) gives something with rational denominator (I am making an assumption that the extension is normal here but shhh).

We are guaranteed that the denominator will be rational by simple Galois theory. The fact that all conjugates of \alpha will then appear on the denominator will give a denominator that is invariant under all elements of the Galois group…so must be rational! (Invariance follows from the fact that left multiplication in a group is an automorphism, so that applying elements of the Galois group will just permute the things in the product).

…so this is why you multiply top and bottom by a-b\sqrt{d}.

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