Polynomial Rings (Units and Zero divisors)


by facepalmer
Tags: divisors, polynomial, rings, units
facepalmer
facepalmer is offline
#1
Feb18-12, 08:43 AM
P: 7
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
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lavinia
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#2
Feb18-12, 09:44 AM
Sci Advisor
P: 1,716
Quote Quote by facepalmer View Post
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?
These three are not units but 2x + 1 is. So is 2x + 3.


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?

Thanks

no but 2x + 2 is a zero divisor.
facepalmer
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#3
Feb19-12, 07:55 AM
P: 7
Thanks, so the constant is required but along with the coefficient of the linear polynomial when determining units and zero-divisors then.

lavinia
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#4
Feb19-12, 09:37 AM
Sci Advisor
P: 1,716

Polynomial Rings (Units and Zero divisors)


Quote Quote by facepalmer View Post
Thanks, so the constant is required but along with the coefficient of the linear polynomial when determining units and zero-divisors then.
no. 2x is also a zero divisor as is 2x^n

But you are right for units.
facepalmer
facepalmer is offline
#5
Feb19-12, 01:56 PM
P: 7
great, thanks for the assistance


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