Primes in Q Fields with Norms less than 6 (Interesting!)

**Samson** · July 27th 2010, 03:33 AM

Hello All,

I've been trying to get the hang of understanding how primes interact with Quadratic fields and then the respective norms. (You can see because over the past month I've had about 6 threads about it).

Anyways, I wanted to examine three Quadratic Fields: Q[Sqrt(-1)], Q[Sqrt(-3)], and Q[Sqrt(-5)].

Can someone identify the primes in each that have a norm less than 6? Once they have been found, how do we determine if they are "associates" of each other?

Thank you! I really appreciate the help!

**undefined** · August 4th 2010, 10:21 AM

Originally Posted by Samson

Hello All,

I've been trying to get the hang of understanding how primes interact with Quadratic fields and then the respective norms. (You can see because over the past month I've had about 6 threads about it).

Anyways, I wanted to examine three Quadratic Fields: Q[Sqrt(-1)], Q[Sqrt(-3)], and Q[Sqrt(-5)].

Can someone identify the primes in each that have a norm less than 6? Once they have been found, how do we determine if they are "associates" of each other?

Thank you! I really appreciate the help!

This may be quite a difficult problem (or might not) and at any rate is certainly not obvious to me. I'm still busy in your other thread(s) regarding these kinds of primes and might have more info when I've done more work in those threads. But for starters, regarding your last question, you can read up on associates and related topics here:

Integral domain - Wikipedia, the free encyclopedia (subheading: Divisibility, prime and irreducible elements)

**Samson** · August 5th 2010, 08:22 AM

Originally Posted by undefined

This may be quite a difficult problem (or might not) and at any rate is certainly not obvious to me. I'm still busy in your other thread(s) regarding these kinds of primes and might have more info when I've done more work in those threads. But for starters, regarding your last question, you can read up on associates and related topics here:

Integral domain - Wikipedia, the free encyclopedia (subheading: Divisibility, prime and irreducible elements)

Thank you for the reference, I will look into it for sure. If you can help after you look at those other threads, that would be great.

Anyone else have any ideas that might be able to help?

**Samson** · August 9th 2010, 04:09 AM

Hey undefined, were you able to dig up anything from those other threads? I really could use some insight on this, and I appreciate the help!

**undefined** · September 4th 2010, 01:32 PM

Originally Posted by Samson

Hey undefined, were you able to dig up anything from those other threads? I really could use some insight on this, and I appreciate the help!

Note: here there is a good amount of rehash of this recent post of mine on a related topic.

Okay so before I was trying to research what primes in a quadratic field $\mathbb{Q}(\sqrt{D})$ were like with respect to that field (since every field is an integral domain, and prime and irreducible elements are defined in integral domains), and unfortunately I missed a very basic piece of number theory and not surprisingly could not dig up much in my research precisely because there are no such primes; every non-zero element in a field is a unit, therefore every non-zero element divides every other element. So when we talk of primes in a quadratic field $\mathbb{Q}(\sqrt{D})$ , what we really mean is the primes in its ring of integers, in particular for this thread, the Gaussian integers $\mathbb{Z}[i]$ , the Eisenstein integers $\mathbb{Z}[\omega]$ where $\omega\equiv\frac{1}{2}(-1+i\sqrt{3})$ , and $\mathcal{O}_{\mathbb{Q}(\sqrt{-5})}$ . (It seems we can alternately define $\mathcal{O}_{\mathbb{Q}(\sqrt{-3})}$ as $\mathbb{Z}[\omega]$ with $\omega = \frac{1}{2}(1+i\sqrt{3})$ ; reference.)

In any case there are several Gaussian primes with small norms listed at MathWorld, likewise here for Eisenstein primes. For more investigation I believe this article on arXiv is directly applicable. I think rather than re-inventing the wheel in this thread and this other one, it would be wise to take the time to read that paper and implement the algorithm, although it could also be fun to try to invent a new one (for primes and/or irreducibles), even if it is unlikely to be as efficient or clever, however my time is limited. I hope you feel satisfied with this response, but do post any further questions.

**Samson** · September 5th 2010, 10:51 AM

Originally Posted by undefined

Note: here there is a good amount of rehash of this recent post of mine on a related topic.

Okay so before I was trying to research what primes in a quadratic field $\mathbb{Q}(\sqrt{D})$ were like with respect to that field (since every field is an integral domain, and prime and irreducible elements are defined in integral domains), and unfortunately I missed a very basic piece of number theory and not surprisingly could not dig up much in my research precisely because there are no such primes; every non-zero element in a field is a unit, therefore every non-zero element divides every other element. So when we talk of primes in a quadratic field $\mathbb{Q}(\sqrt{D})$ , what we really mean is the primes in its ring of integers, in particular for this thread, the Gaussian integers $\mathbb{Z}[i]$ , the Eisenstein integers $\mathbb{Z}[\omega]$ where $\omega\equiv\frac{1}{2}(-1+i\sqrt{3})$ , and $\mathcal{O}_{\mathbb{Q}(\sqrt{-5})}$ . (It seems we can alternately define $\mathcal{O}_{\mathbb{Q}(\sqrt{-3})}$ as $\mathbb{Z}[\omega]$ with $\omega = \frac{1}{2}(1+i\sqrt{3})$ ; reference.)

In any case there are several Gaussian primes with small norms listed at MathWorld, likewise here for Eisenstein primes. For more investigation I believe this article on arXiv is directly applicable. I think rather than re-inventing the wheel in this thread and this other one, it would be wise to take the time to read that paper and implement the algorithm, although it could also be fun to try to invent a new one (for primes and/or irreducibles), even if it is unlikely to be as efficient or clever, however my time is limited. I hope you feel satisfied with this response, but do post any further questions.

When I"m reading this stuff, I"m not reaching any conclusions. Most of this leads me to general solutions, and I'm not sure if that is exactly what I am looking for. I'm pretty sure I should be finding 'finite' answers to this. The original question I posted says:

I wanted to examine three Quadratic Fields: Q[Sqrt(-1)], Q[Sqrt(-3)], and Q[Sqrt(-5)].

Can someone identify the primes in each that have a norm less than 6?

So in a way, I guess I"m looking for the discrete primes in each. Is there a way someone could apply what undefined said (or perhaps undefined) to this and yield discrete primes with norms less than 6 in the fields mentioned above?

**undefined** · September 5th 2010, 11:02 AM

Originally Posted by Samson

When I"m reading this stuff, I"m not reaching any conclusions. Most of this leads me to general solutions, and I'm not sure if that is exactly what I am looking for. I'm pretty sure I should be finding 'finite' answers to this. The original question I posted says:

So in a way, I guess I"m looking for the discrete primes in each. Is there a way someone could apply what undefined said (or perhaps undefined) to this and yield discrete primes with norms less than 6 in the fields mentioned above?

I thought I was quite clear, also what is a "discrete" prime? Quoting from MathWorld on Gaussian primes:

"The primes which are also Gaussian primes are 3, 7, 11, 19, 23, 31, 43, ... (Sloane's A002145). The Gaussian primes with

are given by

,

, 3,

,

."

What's not clear about this?

**Samson** · September 5th 2010, 03:18 PM

What I meant by a discrete prime was actually giving numeric values for primes, not just generics.

Could you separate and specify which ones that you had just posted correspond to which field and confirm that their norms are less than six?

**undefined** · September 5th 2010, 05:01 PM

Originally Posted by Samson

What I meant by a discrete prime was actually giving numeric values for primes, not just generics.

Could you separate and specify which ones that you had just posted correspond to which field and confirm that their norms are less than six?

I understand now you mean "concrete" instead of "discrete". For Gaussian and Eisenstein integers I provided you with links that give small primes. Some of them have norm less than 6, some not, but they are a good list of small primes so I don't know why you would not find them useful. I even quoted them explicitly for Gaussian integers in post #7. In post #5 I provide a link to MathWorld with small primes in Eisenstein integers. Here is the link again.

Eisenstein Prime -- from Wolfram MathWorld

As far as the other field and how to find these elements in general, I have only provided a link to an article that I believe explains the theory and an algorithm to do so. Here is the link

[1001.5214] Primes in quadratic fields

I have not read this in its entirety nor tried to implement it. However I think the article should help you if this is what you really want to learn. I'm a little curious as to why you so strongly would like to be shown how to solve these, step by step from start to finish. Why these particular questions? Is there some use you have in mind for them? Do you see a connection between these primes and some other objects of interest? I'm not here to write a textbook for each individual member. I enjoy helping, but if I wanted to write a textbook, I would do so apart from this. Maybe you need to find a textbook with more worked examples.

Edit: Some more info.

1) You should be able to determine, fully on your own, whether the list of Gaussian primes I gave you includes all Gaussian primes whose norms are less than 6.

2) Here's a nice image of all Gaussian primes with norm less than 500.

http://en.wikipedia.org/wiki/Euclide...ssian_integers

**Samson** · September 7th 2010, 08:26 AM

Originally Posted by undefined

I understand now you mean "concrete" instead of "discrete". For Gaussian and Eisenstein integers I provided you with links that give small primes. Some of them have norm less than 6, some not, but they are a good list of small primes so I don't know why you would not find them useful. I even quoted them explicitly for Gaussian integers in post #7. In post #5 I provide a link to MathWorld with small primes in Eisenstein integers. Here is the link again.

Eisenstein Prime -- from Wolfram MathWorld

As far as the other field and how to find these elements in general, I have only provided a link to an article that I believe explains the theory and an algorithm to do so. Here is the link

[1001.5214] Primes in quadratic fields

I have not read this in its entirety nor tried to implement it. However I think the article should help you if this is what you really want to learn. I'm a little curious as to why you so strongly would like to be shown how to solve these, step by step from start to finish. Why these particular questions? Is there some use you have in mind for them? Do you see a connection between these primes and some other objects of interest? I'm not here to write a textbook for each individual member. I enjoy helping, but if I wanted to write a textbook, I would do so apart from this. Maybe you need to find a textbook with more worked examples.

Edit: Some more info.

1) You should be able to determine, fully on your own, whether the list of Gaussian primes I gave you includes all Gaussian primes whose norms are less than 6.

2) Here's a nice image of all Gaussian primes with norm less than 500.

Euclidean algorithm - Wikipedia, the free encyclopedia

I have chosen these particular questions because they are the first few questions in my "Examples" portion of the text. Once I have been shown properly how to do these in a step-by-step manner, I usually get the rest of them in no time. If you remember, I did post an ad on here for someone that could help full time and I offered to compensate. I'm serious about learning this, I just need someone that can sit down and work through these and show me how its done so I can apply them to the rest of the problem. I started this book in April and I hop to finish it by the end of September if all means possible. Once again, my offer is still on the table.

I looked at both of your links. The one from Wolfram I don't believe specifies which field those are from, nor how to calculate the norm (use the $\omega$ or substitute in what $\omega$ is? Either way, I looked through your document and I read all about those algorithms but I haven't the slightest idea on how to implement them with code or freehand. Some of the operations they used in that are quite high level for me too, probably written by some PhD level mathematician. I didn't see how it mentioning explicitly how to set the field number, nor how to set the search for those concrete primes with norms less than 6.

I apologize if this seems cumbersome, but I am definitely trying to do my best to learn this. Believe me when I say its very frustrating for me as well. I do appreciate your help, and I hope you'll be able to see this through if possible.

**undefined** · September 7th 2010, 09:04 AM

Originally Posted by Samson

I have chosen these particular questions because they are the first few questions in my "Examples" portion of the text. Once I have been shown properly how to do these in a step-by-step manner, I usually get the rest of them in no time. If you remember, I did post an ad on here for someone that could help full time and I offered to compensate. I'm serious about learning this, I just need someone that can sit down and work through these and show me how its done so I can apply them to the rest of the problem. I started this book in April and I hop to finish it by the end of September if all means possible. Once again, my offer is still on the table.

I looked at both of your links. The one from Wolfram I don't believe specifies which field those are from, nor how to calculate the norm (use the $\omega$ or substitute in what $\omega$ is? Either way, I looked through your document and I read all about those algorithms but I haven't the slightest idea on how to implement them with code or freehand. Some of the operations they used in that are quite high level for me too, probably written by some PhD level mathematician. I didn't see how it mentioning explicitly how to set the field number, nor how to set the search for those concrete primes with norms less than 6.

I apologize if this seems cumbersome, but I am definitely trying to do my best to learn this. Believe me when I say its very frustrating for me as well. I do appreciate your help, and I hope you'll be able to see this through if possible.

Well I'd like to just get on with it and do some mathematics, but the issues raised won't go away. I simply don't have time to teach you this much number theory in this much depth, while I'm learning it myself no less, especially since you require so many worked examples in such detail. You consistently ask very basic questions in the midst of trying to solve an advanced problem. So basically I have to help you review grade school algebra in the midst of going over number theory. This is not a very pleasant situation.

"The one from Wolfram I don't believe specifies which field those are from, nor how to calculate the norm"

The Guassian integers are the ring of integers of the field $\mathbb{Q}(\sqrt{-1})$ . This has been noted many times in these threads. The norm for Guassian integers is $N(\alpha)=\alpha\overline{\alpha}=a^2-b^2D=a^2+b^2$ . This is also something you should already know. For the norm in other algebraic integer rings I believe we can use definition 3 in this article on Wikipedia. In the case of Eisenstein integers it reduces to a simple formula, given here under Properties.

Yes the article on arXiv looks rather technical, but believe it or not the questions you ask are rather advanced, so it comes with the territory.

At some point this arrangement probably won't work because I feel there is too much work for me to do, and it's awkward over the internet. Sorry.

Believe me when I say I have put a large amount of time into responding to your threads. And I do wish you the best and will try to help out as time admits, but I'm not committed to making sure you get all these problems worked through step by step as you would like.

**Samson** · September 21st 2010, 08:20 PM

Originally Posted by undefined

I thought I was quite clear, also what is a "discrete" prime? Quoting from MathWorld on Gaussian primes:

"The primes which are also Gaussian primes are 3, 7, 11, 19, 23, 31, 43, ... (Sloane's A002145). The Gaussian primes with

are given by

,

, 3,

,

."

What's not clear about this?

Could you distinguish which ones are part of which Field?

You said these were all of the primes for |a|,|b|<=5, so I take it they a list of all primes within $Q[\sqrt{-1}] , Q[{\sqrt{-3}] , Q[\sqrt{-5}]$ respectively. I could be wrong though in this interpretation.

**Opalg** · September 22nd 2010, 02:43 AM

Originally Posted by Samson

I wanted to examine three Quadratic Fields: Q[Sqrt(-1)], Q[Sqrt(-3)], and Q[Sqrt(-5)].

Can someone identify the primes in each that have a norm less than 6? Once they have been found, how do we determine if they are "associates" of each other?

For the Gaussian integers (the field $\mathbb{Q}[\sqrt{-1}]$ ) the norm is given by $N(a+ib) = a^2+b^2$ . A unit is an element with norm 1, and two elements are associates if you can get one from the other by multiplying it by a unit.

The units in the Gaussian integers are $\pm1$ and $\pm i$ . The only elements with norm less than 6 are the units together with the elements $\pm2$ , $\pm2i$ , $\pm1\pm i$ , $\pm2\pm i$ and $\pm1\pm 2i$ . The elements $\pm2$ , $\pm2i$ are not prime because for example $2 = (1+i)(1-i)$ . All the other elements in that set are prime. There are four units, so each set of associate elements will contain four members. The elements $\pm1\pm i$ form a set of associates. So do the elements $\pm(2+ i)$ and $\pm(1-2i)$ .

In the field $\mathbb{Q}[\sqrt{-3}]$ , the norm is given by $N(a+b\sqrt{-3}) = a^2+3b^2$ . There are only two units, namely $\pm1$ . There are six other elements with norm less than 6, namely $\pm i\sqrt3$ , $\pm2$ and $\pm1\pm i\sqrt3$ . All of these elements are primes. Each element and its negative are associates, but notice that $1+i\sqrt3$ and $1-i\sqrt3$ are not associates.

The analysis for the field $\mathbb{Q}[\sqrt{-5}]$ is similar to that for $\mathbb{Q}[\sqrt{-3}]$ .

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