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Old 2015-06-12, 21:22   #1
wildrabbitt
 
Jul 2014

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Default rational integers

Hi,

have been trying to work out what some examples of rational integers would look like.

I read this article in an attempt to find out :

http://planetmath.org/rationalalgebraicintegers


from the article I'm tempted to think a rational integer is any number n/1 for which n is an ordinary integer.

(but) It seems to me that if 4/1 is a rational integer, then 16/4 is too.

for a set like Z[root(3)] = { x + y(root(3)) : x, y in Z }, a number in this set can be a rational integer iff y = 0.

Any clarifications, help would be much appreciated (if it helps).

Last fiddled with by wildrabbitt on 2015-06-12 at 21:24
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Old 2015-06-12, 21:33   #2
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Quote:
Originally Posted by wildrabbitt View Post
Hi,

have been trying to work out what some examples of rational integers would look like.

I read this article in an attempt to find out :

http://planetmath.org/rationalalgebraicintegers


from the article I'm tempted to think a rational integer is any number n/1 for which n is an ordinary integer.

(but) It seems to me that if 4/1 is a rational integer, then 16/4 is too.

for a set like Z[root(3)] = { x + y(root(3)) : x, y in Z }, a number in this set can be a rational integer iff y = 0.

Any clarifications, help would be much appreciated (if it helps).
reading it a bit I think the proof relies on them being co-prime ( aka not sharing a factor like 16 and 4 do)

Quote:
2∘.  Let the rational number  α=mn  be an algebraic integer where m,n are coprime integers and  n>0.  Then there is a polynomial
so the whole premise of the proof assumes lowest form or proper form.
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Old 2015-06-12, 21:39   #3
Xyzzy
 
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http://mathworld.wolfram.com/RationalInteger.html

Quote:
A synonym for integer. The word "rational" is sometimes used for emphasis to distinguish it from other types of "integers" such as cyclotomic integers, Eisenstein integers, Gaussian integers, and Hamiltonian integers.
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Old 2015-06-13, 00:06   #4
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Quote:
Originally Posted by wildrabbitt View Post
Hi,

have been trying to work out what some examples of rational integers would look like.

I read this article in an attempt to find out :

http://planetmath.org/rationalalgebraicintegers


from the article I'm tempted to think a rational integer is any number n/1 for which n is an ordinary integer.

(but) It seems to me that if 4/1 is a rational integer, then 16/4 is too.

for a set like Z[root(3)] = { x + y(root(3)) : x, y in Z }, a number in this set can be a rational integer iff y = 0.
False. Did you read the definition?

A rational integer is an element of a field whose (algebraic) norm is an element of Z.
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Old 2015-06-13, 00:08   #5
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Originally Posted by Xyzzy View Post
False.
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Old 2015-06-13, 00:35   #6
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In the book Algebraic Number Theory and Fermat's Last Theorem by Ian Stewart and David Tall, the authors state in page 45 that rational integers are the elements of Z, and integers without any qualifier are algebraic integers, i.e., elements of B.

So it is clear that the notation depends on the context.
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Old 2015-06-13, 00:43   #7
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Quote:
Originally Posted by alpertron View Post
In the book Algebraic Number Theory and Fermat's Last Theorem by Ian Stewart and David Tall, the authors state in page 45 that rational integers are the elements of Z, and integers without any qualifier are algebraic integers, i.e., elements of B.

So it is clear that the notation depends on the context.
Agreed. It does depend on context.
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Old 2015-06-13, 00:50   #8
wildrabbitt
 
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Right. Thanks for the replies. Very helpful. To be honest I think I'm out of my depth but I've found this to look at :

[PDF]Introduction to Algebraic Number Theorywww1.spms.ntu.edu.sg/~frederique/ANT10.pdf

by F Oggier - ‎Cited by 2 - ‎Related articles
These are lecture notes for the class on introduction to algebraic number theory, ..... norm of an algebraic integer α in a number field K, based on the different.
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Old 2015-06-13, 00:53   #9
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What is a qualifier for an integer ?
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Old 2015-06-14, 13:21   #10
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An integer is any number whose fractional part is 0.
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Old 2015-06-14, 13:39   #11
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Quote:
Originally Posted by Stargate38 View Post
An integer is any number whose fractional part is 0.
Before Bob gets here I'll remark that your answer is context dependent too. You describe a special case of what mathematicians call an "integer". There are many other kinds of integer which have been defined. For instance, 1+i is a Gaussian integer, and a prime one too. 3 is a prime Gaussian integer too. Two is not prime because 2 = (1+i)*(1-i).
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