Quick Q-Field Problems

Jun 2010
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Hello All,

I had a few quick Quadratic Field related problems I wanted to throw out there to see if someone can give me a hand with.

1 - Does anyone know of a Quadratic Integer in Q[Sqrt(-1)] that is prime but whose norm is not prime?

2 - What is a prime factorization of 6 in Q[Sqrt(-1)] ? I have been having trouble understanding these.

Thank you! -Samson
 
Oct 2009
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Hello All,

I had a few quick Quadratic Field related problems I wanted to throw out there to see if someone can give me a hand with.

1 - Does anyone know of a Quadratic Integer in Q[Sqrt(-1)] that is prime but whose norm is not prime?


If by "quadratic integer" you meant "Gaussian integer" or just "integer", then the only primes in \(\displaystyle \mathbb{Z}\) are elements of the form \(\displaystyle a+bi\) , with \(\displaystyle a^2+b^2=p\,,\,\,p\) an ordinary prime in \(\displaystyle \mathbb{Z}\) , or elements of the form \(\displaystyle a\,\,or\,\,ai\) , with \(\displaystyle |a|=3\!\!\pmod 4\) a rational prime, and the prime \(\displaystyle 1+i\).
It's easy to check all these primes have a prime norm, so...


2 - What is a prime factorization of 6 in Q[Sqrt(-1)] ? I have been having trouble understanding these.


\(\displaystyle 6=2\cdot 3 = -i(1+i)^2\cdot 3=-3i(1+i)^2\) , with \(\displaystyle \pm i\) a unit, of course.

Tonio


Thank you! -Samson

.
 
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Jun 2010
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Hey Tonio, could you edit your post so that I can quote the text that you put inside of my quote?

By the way, thank you on part 2, but what I was going to quote on part 1 was:

By Quadratic Integer, I meant what's described herein : Quadratic integer - Wikipedia, the free encyclopedia

Please note that the article states:
"Quadratic integers are a generalization of the rational integers to quadratic fields. Important examples include the Gaussian integers and the Eisenstein integers."
Can you relate this and the article to find an example to satisfy part one? I know you have the generics, but is there a discrete example you might be able to come up with?
 
Jul 2010
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elements of the form , with a rational prime, and the prime .
It's easy to check all these primes have a prime norm, so...
How exactly do all of them have prime norms? You said there can be prime numbers of the form a (where a is congruent to 3 mod 4). Norm(a) = a^2, and clearly this norm is not a prime. Am I missing something fundamental?
 
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How exactly do all of them have prime norms? You said there can be prime numbers of the form a (where a is congruent to 3 mod 4). Norm(a) = a^2, and clearly this norm is not a prime. Am I missing something fundamental?

Of course you're missing something fundamental: my typos/mistakes.(Giggle)

I wrote that about the prime norms before adding the primes of the form \(\displaystyle a\,,\,ai\,,\,\,with\,\,a=3\!\!\pmod 4\) a prime, and then edited , forgot.

You're right. Thanx

Tonio
 
Jun 2010
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Hey tonio, could you reply to my post above where I attempted to quote you?
 
Oct 2009
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Hey tonio, could you reply to my post above where I attempted to quote you?

I honestly can't understand what you want... can't you read/understand my answer to your post or what?

Tonio
 
Jun 2010
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I honestly can't understand what you want... can't you read/understand my answer to your post or what?

Tonio
Well you never directly answered my last post. I don't know if you missed it. Here is what I said:

Hey Tonio, could you edit your post so that I can quote the text that you put inside of my quote?

By the way, thank you on part 2, but what I was going to quote on part 1 was:

By Quadratic Integer, I meant what's described herein : Quadratic integer - Wikipedia, the free encyclopedia (See Link Above)

Please note that the article states:
"Quadratic integers are a generalization of the rational integers to quadratic fields. Important examples include the Gaussian integers and the Eisenstein integers."
Can you relate this and the article to find an example to satisfy part one? I know you have the generics, but is there a discrete example you might be able to come up with?
 

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Well you never directly answered my last post. I don't know if you missed it. Here is what I said:

Hey Tonio, could you edit your post so that I can quote the text that you put inside of my quote?

By the way, thank you on part 2, but what I was going to quote on part 1 was:

By Quadratic Integer, I meant what's described herein : Quadratic integer - Wikipedia, the free encyclopedia (See Link Above)

Please note that the article states:
"Quadratic integers are a generalization of the rational integers to quadratic fields. Important examples include the Gaussian integers and the Eisenstein integers."
Can you relate this and the article to find an example to satisfy part one? I know you have the generics, but is there a discrete example you might be able to come up with?
EDIT: When I wrote this post originally, I was not careful about the distinction between \(\displaystyle \mathbb{Q}(\sqrt{-1})\) and \(\displaystyle \mathbb{Z}\). Here's what I wrote, slightly edited:

~~~~~~~~~~~~~~~~~~~~~~

tonio's first post (post #2 of this thread) was not "generic" but was a direct and concrete answer to your questions. To spell it out more explicitly,

(1) The answer is, there does not exist any such Gaussian prime.

Further reading:

Gaussian Prime -- from Wolfram MathWorld
Gaussian integer - Wikipedia, the free encyclopedia (subheading: As a unique factorization domain)

Note: I don't know why tonio mentions \(\displaystyle 1+i\) separately, as it fits the description \(\displaystyle a+bi\) where \(\displaystyle N(a+bi)=a^2+b^2=p\) with \(\displaystyle \displaystyle p\) a regular prime in \(\displaystyle \mathbb{Z}\).

(2) tonio gave you the prime factorisation of 6 in \(\displaystyle \mathbb{Z}\). Note that 2*3 is not a prime factorisation here because 2 is not a Gaussian prime.

Further reading:

Unique factorization domain - Wikipedia, the free encyclopedia
Unique Factorization Domain -- from Wolfram MathWorld

~~~~~~~~~~~~~~~~~~~~~~

So, the above applies to \(\displaystyle \mathbb{Z}\). Question to the OP (Samson): Are you sure you want to know about primes in \(\displaystyle \mathbb{Q}(\sqrt{-1})\) as opposed to \(\displaystyle \mathbb{Z}\)? If so, I'll have to think/research more, or someone else will have to add some info..
 
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Oct 2009
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EDIT: When I wrote this post originally, I was not careful about the distinction between \(\displaystyle \mathbb{Q}(\sqrt{-1})\) and \(\displaystyle \mathbb{Z}\). Here's what I wrote, slightly edited:

~~~~~~~~~~~~~~~~~~~~~~

tonio's first post (post #2 of this thread) was not "generic" but was a direct and concrete answer to your questions. To spell it out more explicitly,

(1) The answer is, there does not exist any such Gaussian prime.

Further reading:

Gaussian Prime -- from Wolfram MathWorld
Gaussian integer - Wikipedia, the free encyclopedia (subheading: As a unique factorization domain)

Note: I don't know why tonio mentions \(\displaystyle 1+i\) separately, as it fits the description \(\displaystyle a+bi\) where \(\displaystyle N(a+bi)=a^2+b^2=p\) with \(\displaystyle \displaystyle p\) a regular prime in \(\displaystyle \mathbb{Z}\).



Correct. I was just thinking of primes which are 1 modulo 4 when I wrote \(\displaystyle a^2+b^2=p\) , and tried to distinguish the first prime 2 which isn't 1 modulo 4 and still fulfills the condition: \(\displaystyle 1^2+1^2=2\) , but I didn't write down the distinction.
All the above was thought to convey the fact (theorem) that a prime is a sum of squares iff it is 2 or it is 1 mod 4 , and from here to deduce, together with a little algebra, that an integer is a sum of squares iff any prime equal to 3 mod 4 that appears in its prime decompostion appears there to an even power.
Thanx

Tonio




(2) tonio gave you the prime factorisation of 6 in \(\displaystyle \mathbb{Z}\). Note that 2*3 is not a prime factorisation here because 2 is not a Gaussian prime.

Further reading:

Unique factorization domain - Wikipedia, the free encyclopedia
Unique Factorization Domain -- from Wolfram MathWorld

~~~~~~~~~~~~~~~~~~~~~~

So, the above applies to \(\displaystyle \mathbb{Z}\). Question to the OP (Samson): Are you sure you want to know about primes in \(\displaystyle \mathbb{Q}(\sqrt{-1})\) as opposed to \(\displaystyle \mathbb{Z}\)? If so, I'll have to think/research more, or someone else will have to add some info..

.