- #1
facepalmer
- 7
- 0
Hi all,
I would just like to get some clarity on units and zero-divisors in rings of polynomials.
If I take a ring of Integers, Z4, (integers modulo 4) then I believe the units
are 1 & 3. And the zero-divisor is 2.
Units
1*1 = 1
3*3 = 9 = 1
Zero divisor
2*2 = 4 = 0
Now, If I take a ring of polynomials Z4[x], the polynomials with coefficients in Z4 and wish to find the units I believe that the units in Z4[x] are the constant polynomials 'a' where 'a' in a unit of Z4.
So, 1 and 3.
Now, are the polynomials of degree 1 in Z4[x] with constant values 1 and 3 considered units?
x+1, x+3, 3x+1?
Or are the linear polynomials never considered units? units can only be the constant polynomials?
Does the same apply for the zero-divisors in Z4[x]?
i.e. are the linear polynomials in Z4[x] with constant value 2; x+2, 3x+2, the zero-divisors?
hopefully I am making some sense to this question...
Thanks
I would just like to get some clarity on units and zero-divisors in rings of polynomials.
If I take a ring of Integers, Z4, (integers modulo 4) then I believe the units
are 1 & 3. And the zero-divisor is 2.
Units
1*1 = 1
3*3 = 9 = 1
Zero divisor
2*2 = 4 = 0
Now, If I take a ring of polynomials Z4[x], the polynomials with coefficients in Z4 and wish to find the units I believe that the units in Z4[x] are the constant polynomials 'a' where 'a' in a unit of Z4.
So, 1 and 3.
Now, are the polynomials of degree 1 in Z4[x] with constant values 1 and 3 considered units?
x+1, x+3, 3x+1?
Or are the linear polynomials never considered units? units can only be the constant polynomials?
Does the same apply for the zero-divisors in Z4[x]?
i.e. are the linear polynomials in Z4[x] with constant value 2; x+2, 3x+2, the zero-divisors?
hopefully I am making some sense to this question...
Thanks