# Euclidean domains

• March 9th 2010, 10:25 AM
choo
Euclidean domains
Hi,

i need to show which of the following are euclidean domains:

a) C[x] with d(f)=deg(f) (where C[x] = set of polynomials with complex coefficients)
b) set of integers with d(n)=|n|+1
c) ring of integers modulo 5 with d(f)=2deg(f)
d) ring of integers modulo 6 with d(f)=deg(f)
e) Z(w) where w=(1+root(-3))/2, with d(z)=|z|^2

i know that an integral domain is a Euclidean domain if it has a Euclidean function but i dont know how to start off the question.

Thanks!
• March 9th 2010, 01:48 PM
Drexel28
Quote:

Originally Posted by choo
Hi,

i need to show which of the following are euclidean domains:

a) C[x] with d(f)=deg(f) (where C[x] = set of polynomials with complex coefficients)
b) set of integers with d(n)=|n|+1
c) ring of integers modulo 5 with d(f)=2deg(f)
d) ring of integers modulo 6 with d(f)=deg(f)
e) Z(w) where w=(1+root(-3))/2, with d(z)=|z|^2

i know that an integral domain is a Euclidean domain if it has a Euclidean function but i dont know how to start off the question.

Thanks!

these are all mostly plug and chug. I'll help with the first one.

We may assume WLOG that $p,p'\ne 0$. Thus, $\deg(pp')=\deg(p)+\deg(p')\geqslant deg(p)$.

The other bit follows from this.

Try the others and get back to us.
• March 13th 2010, 05:50 AM
choo
for part b) this is what i got:

the function is strictly increasing.
a=bq+r
|r| <= |b/2|
this means that d(b/2) < d(b) ?
• March 13th 2010, 10:34 PM
Drexel28
Quote:

Originally Posted by choo
for part b) this is what i got:

the function is strictly increasing.
a=bq+r
|r| <= |b/2|
this means that d(b/2) < d(b) ?

What does that matter? You want to prove that $d(a)\leqslant d(ab)$ for every $a,b\in \mathbb{Z}-\{0\}$, right?

So $d(ab)=|ab|+1=|a||b|+1$ now since $b\in\mathbb{Z}$ we know that $|b|\geqslant 1$ and so $|a||b|+1\geqslant |a|+1=d(a)$
• March 14th 2010, 04:28 AM
choo
thanks for ur help. heres what i did for the other parts:

c) d(ab)=2deg(ab), b>=[1] and so
2deg(ab)= 2(deg(a)+deg(b))>=2deg(a)=d(a)

d) the ring of integers modulo 6 is not a field.
d(ab)=deg(ab)=deg(a)+deg(b)>=deg(a)

e) d(w)=|w|^2 = 1/4+3/4=1
(what do i do next?)

thanks