The Tower Rule
October 13, 2011 1 Comment
Yesterday, we noticed that . We can use a similar argument to deduce that . What does that tell us about ? You may have guessed that . Now you’re either thinking that or . Those of you thinking that are correct!
In general, if I have fields , , and , with , I can think of as a vector space over the field and as a vector space over . What’s more, I can think of as a vector space over the field . Then I can look at the dimensions, and get what I declare to be the coolest result in general field theory:
Remember, if you want, you can think about this as , but the former way I think looks nicer.
Let be a basis for the vector space over the field . Let be a basis for the vector space over the field . We want to find a basis for the vector space over the field , and we want it to have elements. There’s really only one logical choice, so let’s hope it works. We’ll show that is a basis for over the field .
So take . We know there is a unique way to write it where . But then each of these can be written uniquely as . So we have
We can think of this once again as a vector space over the field , and let , giving us
But is a basis for the vector space over the field , and hence a linearly independent set, so it must be that for every , . Now we have is a basis and hence linearly independent, so each of the must be zero, for any or . This is exactly what it means for to be linearly independent. WOOHOO!
If you didn’t think that was cool, you have no business calling yourself a decent human being.