Renumberings and algebraic integers

Hi,

Many thanks for the revision class last week. I just have a few more specific questions I wondered if you would be able to help me with.

1) You often refer to ‘Theorem 107’ in solutions however this doesn’t seem to correlate with the notes I have, is there any chance you could clarify what this theorem says and how it is applied?

2) In ‘An example of an integral basis’ you select y\sigma_3(y), is there a reason for this rather than y\sigma_2(y)?

Many thanks for your help.

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Reply:

I’m sorry. When I revised the notes, I forgot to update the references on the supplementary material. Most of the time, I do think it’s a good exercise to locate the correct reference somewhere near the incorrect one. (This is not just an excuse for me to be lazy!) In the case of the old theorem 107, this is theorem 111 in the current notes. It allows us to rule out primes p from the possible denominators of algebraic integers expressed in terms of a basis B=\{1,  \alpha, \alpha^2,  \ldots ,\alpha^d\} when the minimal polynomial for the algebraic integer \alpha is Eisenstein for the prime p.

Regarding the second question, I used y\sigma_3(y) because I tried it out and it worked. It’s a good exercise to try the argument with y\sigma_3(y) instead and see if it’s just as good. It shouldn’t be immediately obvious until you’ve followed the pattern of the given proof a good deal of the way. Let me know how it goes.

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