20150612, 21:22  #1 
Jul 2014
447_{10} Posts 
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 20150612 at 21:24 
20150612, 21:33  #2  
"Forget I exist"
Jul 2009
Dumbassville
8384_{10} Posts 
Quote:
Quote:


20150612, 21:39  #3  
Aug 2002
210B_{16} Posts 
http://mathworld.wolfram.com/RationalInteger.html
Quote:


20150613, 00:06  #4  
Nov 2003
2^{2}×5×373 Posts 
Quote:
A rational integer is an element of a field whose (algebraic) norm is an element of Z. 

20150613, 00:08  #5  
Nov 2003
1110100100100_{2} Posts 
Quote:


20150613, 00:35  #6 
Aug 2002
Buenos Aires, Argentina
1,447 Posts 
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. 
20150613, 00:43  #7  
Nov 2003
1D24_{16} Posts 
Quote:


20150613, 00:50  #8 
Jul 2014
3×149 Posts 
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. 
20150613, 00:53  #9 
Jul 2014
3×149 Posts 
What is a qualifier for an integer ?

20150614, 13:21  #10 
"Daniel Jackson"
May 2011
14285714285714285714
2·3^{3}·13 Posts 
An integer is any number whose fractional part is 0.

20150614, 13:39  #11 
Bamboozled!
"𒉺𒌌𒇷𒆷𒀭"
May 2003
Down not across
3·7^{3}·11 Posts 
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)*(1i).

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