Find all the monic irreducible polynomials

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In summary, a monic irreducible polynomial is a polynomial with a leading coefficient of 1 and cannot be factored into polynomials of lower degree with coefficients in the same field. To find all the monic irreducible polynomials of a certain degree, Eisenstein's criterion or Berlekamp's algorithm can be used. These methods involve checking the divisibility of the polynomial by certain prime numbers. Monic irreducible polynomials cannot have multiple roots, and finding all of them is important in areas of mathematics such as algebraic number theory and algebraic geometry. There are an infinite number of monic irreducible polynomials since there are infinitely many prime numbers.
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Homework Statement



Find all the monic irreducible polynomials of degree [tex]\leq 3[/tex] in [tex]\mathbb{F}_2[x][/tex], and the same in [tex]\mathbb{F}_3[x][/tex].

Homework Equations


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The Attempt at a Solution


n/a
 
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What can you say about the irreducible factors of [itex]x^{p^n} - x \in \mathbb{F}_p[x][/itex]?
 

1. What is the definition of a monic irreducible polynomial?

A monic irreducible polynomial is a polynomial with a leading coefficient of 1 and cannot be factored into polynomials of lower degree with coefficients in the same field.

2. How do you find all the monic irreducible polynomials of a certain degree?

To find all the monic irreducible polynomials of a certain degree, you can use the Eisenstein's criterion or the Berlekamp's algorithm. These methods involve checking the divisibility of the polynomial by certain prime numbers and can be done by hand or using a computer program.

3. Can monic irreducible polynomials have multiple roots?

No, monic irreducible polynomials cannot have multiple roots. This is because if a polynomial has a multiple root, it can be factored into lower degree polynomials, violating the definition of irreducibility.

4. What is the significance of finding all the monic irreducible polynomials?

Finding all the monic irreducible polynomials is important in many areas of mathematics, such as algebraic number theory and algebraic geometry. These polynomials have unique properties that make them useful in solving equations and studying the structure of mathematical objects.

5. Are there an infinite number of monic irreducible polynomials?

Yes, there are an infinite number of monic irreducible polynomials since there are infinitely many prime numbers. Each prime number corresponds to a different monic irreducible polynomial, so the set of monic irreducible polynomials is infinite.

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