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 Register FAQ Search Today's Posts Mark Forums Read  2015-06-12, 21:22 #1 wildrabbitt   Jul 2014 44710 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 2015-06-12 at 21:24   2015-06-12, 21:33   #2
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

838410 Posts Quote:
 Originally Posted by wildrabbitt 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.   2015-06-12, 21:39   #3
Xyzzy

Aug 2002

210B16 Posts 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.    2015-06-13, 00:06   #4
R.D. Silverman

Nov 2003

22×5×373 Posts Quote:
 Originally Posted by wildrabbitt 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.   2015-06-13, 00:08   #5
R.D. Silverman

Nov 2003

11101001001002 Posts Quote:
 Originally Posted by Xyzzy http://mathworld.wolfram.com/RationalInteger.html False.   2015-06-13, 00:35 #6 alpertron   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.   2015-06-13, 00:43   #7
R.D. Silverman

Nov 2003

1D2416 Posts Quote:
 Originally Posted by alpertron 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.   2015-06-13, 00:50 #8 wildrabbitt   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.   2015-06-13, 00:53 #9 wildrabbitt   Jul 2014 3×149 Posts What is a qualifier for an integer ?   2015-06-14, 13:21 #10 Stargate38   "Daniel Jackson" May 2011 14285714285714285714 2·33·13 Posts An integer is any number whose fractional part is 0.   2015-06-14, 13:39   #11
xilman
Bamboozled!

"𒉺𒌌𒇷𒆷𒀭"
May 2003
Down not across

3·73·11 Posts Quote:
 Originally Posted by Stargate38 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).   Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post MattcAnderson MattcAnderson 2 2016-12-14 16:58 Joshua2 Puzzles 19 2009-11-08 00:36 jinydu Lounge 4 2008-10-01 07:45 nibble4bits Math 5 2008-01-08 04:58 mfgoode Homework Help 9 2007-08-19 07:19

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