How Do You Determine Irreducible Polynomials Over Finite Fields?

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In summary, the conversation discusses finding all irreducible polynomials of the form x^2 + ax + b in the field \mathbb{F}_3 with 3 elements. It also demonstrates that \mathbb{F}_3(x)/(x^2 + x + 2) is a field by computing its multiplicative monoid. Lastly, it identifies [\mathbb{F}_3(x)/(x^2 + x + 2)]* as an abstract group and suggests using brute force to solve the problem.
  • #1
mathusers
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(1):
Find all irreducible polynomials of the form [itex]x^2 + ax +b [/itex], where a,b belong to the field [itex]\mathbb{F}_3[/itex] with 3 elements.
Show explicitly that [itex]\mathbb{F}_3(x)/(x^2 + x + 2)[/itex] is a field by computing its multiplicative monoid.
Identify [[itex]\mathbb{F}_3(x)/(x^2 + x + 2)[/itex]]* as an abstract group.

any suggestions please?
 
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  • #2
no, but I am currently doing problems that look a lot like this. I would really enjoy seeing this problem solved. =).
 
  • #3
mathusers said:
(1):
Find all irreducible polynomials of the form [itex]x^2 + ax +b [/itex], where a,b belong to the field [itex]\mathbb{F}_3[/itex] with 3 elements.
Show explicitly that [itex]\mathbb{F}_3(x)/(x^2 + x + 2)[/itex] is a field by computing its multiplicative monoid.
Identify [[itex]\mathbb{F}_3(x)/(x^2 + x + 2)[/itex]]* as an abstract group.

any suggestions please?
It's a very small problem. Have you tried brute force?
 

FAQ: How Do You Determine Irreducible Polynomials Over Finite Fields?

What is an irreducible polynomial?

An irreducible polynomial is a polynomial that cannot be factored into polynomials of lower degree with coefficients in the same field. In other words, it cannot be broken down into simpler components.

How do you determine if a polynomial is irreducible?

There are a few methods for determining if a polynomial is irreducible, such as using the rational root theorem or checking for irreducibility criteria such as Eisenstein's criterion. In general, it involves checking for factors of lower degree and making sure they do not exist.

Why are irreducible polynomials important?

Irreducible polynomials are important in many areas of mathematics, such as in algebraic number theory and algebraic geometry. They also play a crucial role in coding theory, as they are used to generate finite fields which are used in error-correcting codes.

Can an irreducible polynomial have multiple roots?

No, an irreducible polynomial cannot have multiple roots. This is a consequence of the fundamental theorem of algebra, which states that a polynomial of degree n can have at most n distinct roots.

Are irreducible polynomials unique?

No, there can be multiple irreducible polynomials of the same degree with different coefficients. For example, x^2 + 1 and x^2 + 2 are both irreducible polynomials of degree 2.

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