'Let y be a root of f(t) = t^3+0.5t+1 (irreducible/Q)
Then clearly y is not an algebraic integer.'
Can I ask why this is? I mean why is it 'clearly'? Is it not possible to have a polynomial g with integer coefficients st g(y) = 0?
Thanks

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 24032013 22:27

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 24032013 23:02
(Original post by 2710)
'Let y be a root of f(t) = t^3+0.5t+1 (irreducible/Q)
Then clearly y is not an algebraic integer.'
Can I ask why this is? I mean why is it 'clearly'? Is it not possible to have a polynomial g with integer coefficients st g(y) = 0?
ThanksLast edited by Indeterminate; 24032013 at 23:03. 
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 24032013 23:10
(Original post by Indeterminate)
By definition, an algebraic integer is a root of a monic polynomial with integer coefficients. Monic means that the coefficient of the term with the highest power is 1. This is a monic polynomial, but 0.5 isn't an integer.
I know obviously that the above eqn does not have integer coefficients. But thats not what im asking. If you read my question, im asking, why can there not be a different eqn somewhere out there, with integer coefficients st f(y)=0? 
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 24032013 23:20
(Original post by 2710)
An element y is considered an algebraic integer if there exists a monic polynomial f(t) e Z[t] st f(y) = 0
I know obviously that the above eqn does not have integer coefficients. But thats not what im asking. If you read my question, im asking, why can there not be a different eqn somewhere out there, with integer coefficients st f(y)=0? 
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 24032013 23:21
(Original post by 2710)
An element y is considered an algebraic integer if there exists a monic polynomial f(t) e Z[t] st f(y) = 0
I know obviously that the above eqn does not have integer coefficients. But thats not what im asking. If you read my question, im asking, why can there not be a different eqn somewhere out there, with integer coefficients st f(y)=0?
EDIT: Since we're given that it's irreducible over Q, we know what the minimal polynomial is, which makes things a lot more straightforward.Last edited by Mark13; 24032013 at 23:25. 
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 24032013 23:23
(Original post by Indeterminate)
This is because y is already a root of a polynomial that's not monic, hence it cannot be an algebraic integer by definition. 
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 24032013 23:34
(Original post by Mark13)
The definition of an algebraic integer is a complex number that satisfies a monic polynomial with integer coefficients  algebraic integers can (and do) satisfy nonmonic polynomials. Anyway, t^3+t/2+1 is monic...
What is to say that there isnt a polynomial g st:
(t^3+t/2+1)g = monic with integer coefficients?
Because if there is one, then y would staisfy this eqn, it is monic, and also has integer coefficients. I know there isnt one, coz I know the answer, but I just want to know why there isnt one? 
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 24032013 23:37
(Original post by 2710)
Ok, so this is the minimal polynomial.
What is to say that there isnt a polynomial g st:
(t^3+t/2+1)g = monic with integer coefficients?
Because if there is one, then y would staisfy this eqn, it is monic, and also has integer coefficients. I know there isnt one, coz I know the answer, but I just want to know why there isnt one? 
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 24032013 23:42
(Original post by Mark13)
See the edit to my previous post: you know the minimal polynomial of y must divide t^3+t/2+1, but you also know that t^3+t/2+1 is irreducible over Q, which tells you... 
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 24032013 23:45
(Original post by 2710)
nono, you cannot use that reasoning. because t^3+t/2+1 IS the minimal polynomial. Right? And then the question in my post before standsLast edited by Mark13; 24032013 at 23:59. 
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 24032013 23:47
(Original post by Mark13)
It's a standard fact that a complex number is an algebraic integer iff its minimal polynomial has coefficients in Z. Go ahead and prove it if you want. 
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 24032013 23:59
(Original post by 2710)
The definition im reading does not say anything about minimal 
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 25032013 00:01
(Original post by Mark13)
Whatever definition you're using, it's a standard result that a complex number is an algebraic integer if and only if its minimal polynomial over Q has coefficients in Z. 
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 25032013 00:01
(Original post by Mark13)
The definition of an algebraic integer is a complex number that satisfies a monic polynomial with integer coefficients  algebraic integers can (and do) satisfy nonmonic polynomials. Anyway, t^3+t/2+1 is monic...
i meant to say that it doesn't have integer coefficients but that makes no difference I guess 
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 25032013 00:04
(Original post by Mark13)
Whatever definition you're using, it's a standard result that a complex number is an algebraic integer if and only if its minimal polynomial over Q has coefficients in Z. 
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 25032013 00:04
Pulled straight from that page:
"The following are equivalent definitions of an algebraic integer...
\alpha \in K is an algebraic integer if there exists a monic polynomial f(x) \in \mathbb{Z}[x] such that f(\alpha) = 0.
\alpha \in K is an algebraic integer if the minimal monic polynomial of \alpha over \mathbb Q is in \mathbb{Z}[x]." 
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 25032013 00:07
(Original post by Mark13)
Pulled straight from that page:
"The following are equivalent definitions of an algebraic integer...
\alpha \in K is an algebraic integer if there exists a monic polynomial f(x) \in \mathbb{Z}[x] such that f(\alpha) = 0.
\alpha \in K is an algebraic integer if the minimal monic polynomial of \alpha over \mathbb Q is in \mathbb{Z}[x]."
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