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# Invertible elements proof watch

1. Hi, I've done the proof for the question reading "prove that x element of R, is an invert know element IF N(x)=1. And just wanted to know if I had went wrong anywhere! Also I'm stuck on the last part, the deducing the invertible elements ? Be great if anyone could help thanks! Attachment 511063

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2. @Zacken

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3. @Bath_Student say again?

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4. (Original post by maths10101)
...
Heya! Your tag didn't work so I didn't get a notification and this is in the wrong forum (should be in maths, in the future stick to posting questions in the maths forum), sorry for not seeing it sooner. I'll have a look at your question now.
5. (Original post by maths10101)
...
Okay, let's do the deduce bit. The only invertible elements of the ring are going to satisfy where is the inverse to . Then you know that from the fact that , yeah?

Now try writing z = a+ib and w = c+ id and see if you can take it from there. If not:

more hints
This is the same thing as saying since and .

What does this imply about and ?

More hint
You know that (because ring), . So and are both non-negative integers which must both equal 1 since they need to multiply to 1 in the integers.

You know that , so... what does that mean for and ?

More hints
So you need which means that you should be able to easily check your solutions are of the form or...? I'll let you fill in the blanks.
6. (Original post by Zacken)
Okay, let's do the deduce bit. The only invertible elements of the ring are going to satisfy where is the inverse to . Then you know that from the fact that , yeah?

Now try writing z = a+ib and w = c+ id and see if you can take it from there. If not:

more hints
This is the same thing as saying since and .

What does this imply about and ?

More hint
You know that (because ring), . So and are both non-negative integers which must both equal 1 since they need to multiply to 1 in the integers.

You know that , so... what does that mean for and ?

More hints
So you need which means that you should be able to easily check your solutions are of the form or...? I'll let you fill in the blanks.
A lot of thanks for this help mate!..I'll look into it in a second to ensure I am okay with it...however in terms of the picture I uploaded, is the method/what I've done, look okay to show N(X)=1?

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@Zacken
7. (Original post by maths10101)
A lot of thanks for this help mate!..I'll look into it in a second to ensure I am okay with it...however in terms of the picture I uploaded, is the method/what I've done, look okay to show N(X)=1?

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@Zacken

I think your method is okay.
8. (Original post by Zacken)
Okay, let's do the deduce bit. The only invertible elements of the ring are going to satisfy where is the inverse to . Then you know that from the fact that , yeah?

Now try writing z = a+ib and w = c+ id and see if you can take it from there. If not:

more hints
This is the same thing as saying since and .

What does this imply about and ?

More hint
You know that (because ring), . So and are both non-negative integers which must both equal 1 since they need to multiply to 1 in the integers.

You know that , so... what does that mean for and ?

More hints
So you need which means that you should be able to easily check your solutions are of the form or...? I'll let you fill in the blanks.
Alrite, so the blanks would just be
"of the form (+-1,0) or (0,+-i). correct?
or is there more of this proof?
9. (Original post by maths10101)
Alrite, so the blanks would just be
"of the form (+-1,0) or (0,+-i). correct?
or is there more of this proof?
Nopes, that's pretty much it - methinks.

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