Find All Ring Homomorphisms f: Z[√2] → Z 7

[andy1224]

I need to find all the ring homomorphisms of f:Z[sqrt(2)]->Z 7

basically I don't even know where to start. any suggestions would be great

 

[matt grime]

Hmm. Yes, *you* need to find them, and *you* need to tell us what *you* have done, and *you* need to stop posting the same question in more than one place.

You start, as always, with the definitions of the objects in question. So what are they, and what is a homomorphism?

 

[andy1224]

wow.
1- yes *I* do need to find them so I'm not really sure about why you enjoy using the * key so much but yeah..
2- its my first time using this so I'm sincerely sorry that it's in two places
3- I didn't give the definition of a homormorphism since not only is it pretty common for anyone who has done group/ring theory but I also figured that if someone didn't know what a homomorphism is then they would most likely not be able to help me in the first place
4- there aren't really definitions of Z[sqrt(2)], again you either know what it is or you don't depending on how much group/ring theory you've done.
5- I haven't started it which is why I said "I don't know where to start" or else I would've showed what I've done.

so yeah thanks for your informative post. I just didn't know how to start the problem. I wasn't looking for just the answer- but thanks for assuming I was.

 

[matt grime]

Asking you what the definition is is important. What others know isn't important. It is what you know. So put down the definitions. (This also serves to make sure you have the correct things in mind, and are attempting to prove what you actually need to prove).

If you start from the definitions, you will get the answers.

So, what are the definitions?

Z[sqrt(2)] = a+b*sqrt(2) for a, b in Z. What do you need to do to specify where a homomorphism sends any element of Z[sqrt(2)]?

Try to find something that satsifies the definition of a homomorphism. The definitions are the place to start. You might want to explain what you mean by Z 7 as well, since that doesn't make any sense. Do you mean Z/7Z, or Z_(7) or Z_7 (_ is ascii for subscript)? Those are all (to different people) different things: the integers mod 7 the 7-locals and the 7-adics.