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Let be a finite integral domain that is not a field. For some there is no for some many as is not a field. Consider the powers of ,
let,
Consider the first powers as unique, implying that is not unique or consider the first as not unique. In any given case for some ,
If that implies that (this can be shown inductively) and we know that so that implies that,
Although that implies that has an inverse, that begins the only numbers that could not have an inverse indeed do have an inverse thus all numbers in have an inverse thus is a field!
let,
Consider the first powers as unique, implying that is not unique or consider the first as not unique. In any given case for some ,
If that implies that (this can be shown inductively) and we know that so that implies that,
Although that implies that has an inverse, that begins the only numbers that could not have an inverse indeed do have an inverse thus all numbers in have an inverse thus is a field!
This post has been edited 1 time. Last edited by Thayaden, Nov 7, 2024, 7:43 PM