Finding the units in the ring F[x]

  • MHB
  • Thread starter hmmmmm
  • Start date
  • Tags
    Ring Units
In summary, a ring in mathematics is a set of elements with two binary operations that follow certain rules and properties. Units in a ring are elements that have a multiplicative inverse, which can be found by multiplying each element with every other element in the ring or by using the Euclidean algorithm. A ring can have more than one unit, with the identity element and its additive inverse being the minimum. In the ring F[x], the units are all polynomials with a non-zero constant term.
  • #1
hmmmmm
28
0
I have to find all the units in the ring $F[x]$ where $F$ is a field.

Clearly all polynomials of degree 0 are units as they are in the field F.

Now suppose $\alpha(x)\beta(x)=1$ which gives $\mbox{deg}(\alpha(x))=-\mbox{deg}(\beta(x))$ which gives $\mbox{deg}(\beta(x)=\mbox{deg}(\alpha(x)=0$ so the units are precisely the polynomials of degree.

Is this correct?

Thanks for any help
 
Last edited:
Physics news on Phys.org
  • #2
Re: Finding the units in the ring $F[x]$

That is correct. The units are the zero-degree polynomials (the nonzero elements of the field).
 

FAQ: Finding the units in the ring F[x]

What is a ring in mathematics?

A ring in mathematics is a set of elements with two binary operations, usually addition and multiplication, that follow certain rules and properties. It is a fundamental concept in abstract algebra and is used to study algebraic structures.

What are units in a ring?

In a ring, a unit is an element that has a multiplicative inverse. This means that when multiplied by another element, it results in the identity element, which is usually denoted as 1. In other words, a unit is an element that has a reciprocal that is also in the ring.

How do you find units in a ring?

To find units in a ring, you need to check if each element has a multiplicative inverse. This can be done by multiplying each element with every other element in the ring and seeing if it results in the identity element. If it does, then that element is a unit. Additionally, in a finite ring, you can use the Euclidean algorithm to find the multiplicative inverse of an element.

Can a ring have more than one unit?

Yes, a ring can have more than one unit. In fact, every ring has at least two units - the identity element and its additive inverse. However, in some rings, there can be multiple elements that have multiplicative inverses, and these are also considered units.

How do you determine the units in the ring F[x]?

In the ring F[x], where F is a field, the units are all the polynomials with a non-zero constant term. This is because in a field, every non-zero element has a multiplicative inverse, and in the ring F[x], the constant term is the only non-zero term in a polynomial of degree 0.

Similar threads

I Splitting ring of polynomials - why is this result unfindable?
Replies
9
Views
2K
MHB The irreducible polynomial is not separable
Replies
1
Views
2K
I Algebraic Extensions - Dummit and Foote, Propn 11 and 12 ....
Replies
3
Views
2K
MHB Algebraic Extensions - Dummit and Foote, Propositions 11 and 12 .... ....
Replies
2
Views
1K
Division Algorithm for Polynomials in R[x] confusing me
Replies
1
Views
2K
I Proving properties of polynomial in K[x]
Replies
8
Views
1K
MHB Ideals in Polynomial rings - Knapp - page 146
Replies
4
Views
1K
MHB Zero divisor for polynomial rings
Replies
1
Views
2K
MHB Rotman - Theorem 2.64 - Irreducibility
Replies
4
Views
1K
MHB Field Theory - Algebrais Extensions - Dummit and Foote Section 13.2
Replies
4
Views
2K

Hot Threads

Recent Insights

Back
Top