What Are the Units in the Ring F[x] for a Field F?

Thread starter sillyquestions
Start date Oct 14, 2005
Tags

Field Units

In summary, a unit of F[x] in a Field F is an element that has a multiplicative inverse in the field. It can be determined if an element is a unit by checking if it is non-zero and has no common divisors with any other non-zero element in the field. Units in F[x] are significant in determining the structure and properties of the field, as well as in solving equations and finding inverses. An element in F[x] can be a unit in one field but not in another, as the set of units depends on the specific field and its properties. Units and irreducible polynomials in F[x] are closely related, as an element is a unit if and only if it is not divisible by any

Oct 14, 2005

sillyquestions

If F is a field obtain all the units in F[x]?

Physics news on Phys.org

Oct 14, 2005

Hurkyl

Staff Emeritus

Science Advisor

Gold Member

Well, this sounds like homework... so what have you tried on this problem? Have you at least come up with any ideas on how to approach it, no matter how stupid it may seem?

Oct 14, 2005

HallsofIvy

Science Advisor

Homework Helper

In particular: What is the definition of "Field" and what is the definition of "unit"?

FAQ: What Are the Units in the Ring F[x] for a Field F?

What is a unit of F[x] in a Field F?

A unit of F[x] in a Field F is an element that has a multiplicative inverse in the field. In other words, it can be multiplied by another element to produce the multiplicative identity, which is typically denoted as 1.

How do you determine if an element is a unit in F[x] in a Field F?

An element in F[x] is a unit if and only if it is a non-zero element and its greatest common divisor with the polynomial 1 is equal to 1. This can also be expressed as having no common divisors with any other non-zero element in the field.

What is the significance of units in F[x] in a Field F?

Units in F[x] play a crucial role in determining the structure and properties of the field. They are also important in solving equations and finding inverses of elements.

Can an element in F[x] be a unit in one field but not in another?

Yes, an element in F[x] can be a unit in one field but not in another. This is because the set of units in a field depends on the specific field and its properties.

How are units related to irreducible polynomials in F[x] in a Field F?

Units and irreducible polynomials in F[x] are closely related. In fact, an element in F[x] is a unit if and only if it is not divisible by any irreducible polynomial in F[x]. Additionally, every non-zero element in F[x] can be represented as a product of a unit and an irreducible polynomial.