Thread: checking answer on units of integer

U(Z_4) ={ 1, 3}. Z_4 is integers mod n
In general, U(Z_n) = { a in Z_n: hcf(a,n) =1}
And
U(Z_4[X]) = { 0, 1, 2 ,3}. Is this correct?
where Z_n[X] is polynomial ring over Z generated by one variable
How do you generate the formula of U(Z_n[X])?

Thank you

Originally Posted by knguyen2005

U(Z_4) ={ 1, 3}. Z_4 is integers mod n
In general, U(Z_n) = { a in Z_n: hcf(a,n) =1}
And
U(Z_4[X]) = { 0, 1, 2 ,3}. Is this correct?


Nop. In general, , when R is a ring, so in this case the units of the polynomial ring are the same as the ring's.

Tonio

where Z_n[X] is polynomial ring over Z generated by one variable
How do you generate the formula of U(Z_n[X])?

Thank you

.

Thank you Tonio, but I think you misunderstood my question
I know in general but what I meant was how do you write down the elements of the set
Example: U(Z_4)={1, 3} since these elements of the set are coprime to 4
Ok, how about the polynomial of the ring over Z_4. Can we find the elements in U(Z_4[x]) ?
is it U(Z_4[X]) = { 0, 1, 2 ,3} or U(Z_4[X]) = { 1, 2 ,3} ?

if is a commutative ring with identity element, then holds only for reduced rings, i.e. rings with no non-zero nilpotent elements.

in general if and only if and is nilpotent for all see my post in here: http://www.mathhelpforum.com/math-he...ial-rings.html

for example if then if and only if and