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 May 8th, 2017, 06:23 PM #1 Banned Camp   Joined: Apr 2016 From: Australia Posts: 244 Thanks: 29 Math Focus: horses,cash me outside how bow dah, trash doves An example of a non rational algebraic integer Hi I was hoping someone could give me an example of a non rational algebraic integer please
May 8th, 2017, 07:49 PM   #2
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 Originally Posted by Adam Ledger Hi I was hoping someone could give me an example of a non rational algebraic integer please
According to Wiki, $\sqrt{5}$ is an algebraic integer and of course it's not rational. It's a root of $x^2 - 5$.

Also here is a nice explanation. They give the example that $4 + \sqrt{7}$ is an algebraic integer but $\frac{4 + \sqrt{7}}{2}$ isn't. I didn't work out the details but offhand it looks like the latter is a root of a polynomial like $2x^2$-something which is against the rules for algebraic integers.

The algebraic integers are the roots (in the complex numbers, which includes the reals) of all the monic polynomials with integer coefficients. Monic means the leading coefficient is $1$.

 May 9th, 2017, 11:29 PM #3 Banned Camp   Joined: Apr 2016 From: Australia Posts: 244 Thanks: 29 Math Focus: horses,cash me outside how bow dah, trash doves Thanks, so Eisenstein integers https://en.wikipedia.org/wiki/Eisenstein_integer would be a sub-ring of this ring if I'm correct, my basic goal for this exercise is to find the neatest clear-cut means of defining the entire ring, which I'm guessing would just be a set declaring the necessity of all of its elements being roots of a monic poly with integer coefficients. Thanks anyway, it just case of brain freeze having to readapt my definition of an integer to include those on the complex plane! Last edited by skipjack; May 9th, 2017 at 11:58 PM.

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