What is the Probability of Factoring Random Monic Polynomials?

  • Thread starter lttlbbygurl
  • Start date
  • Tags
    Polynomials
In summary, the conversation discusses finding the probability that a random monic polynomial over F_3 of degree exactly 10 will factor into a product of polynomials of degree less than or equal to 2. It also mentions the probability for a random monic polynomial of degree at most 10 to factor in such a way. The conversation also considers the number of different monic polynomials of degree 10, as well as the number of monic polynomials of degree 1 and irreducible monic polynomials of degree 2. Finally, it addresses the different ways these polynomials can be multiplied to get a polynomial of degree 10.
  • #1
lttlbbygurl
6
0
I've been trying to work out a bunch of problems that have to do with finding irreducible polynomials, and this one really seemed to stump me...

What is the probability that a random monic polynomial over [tex] F_3 [/tex] of degree exactly 10 factors into a product of polynomials of degree less than or equal to 2? What is the probability that a random monic polynomial of degree at most 10 factors into such a product?
 
Physics news on Phys.org
  • #2
Let's do monic polynomials. (Multiply them by 2 to get the other polynomials, and similarly for the other factorizations, so the probability answer is the same.)

How many different monic polynomials of degree exactly 10 are there?

How many different monic polynomials of degree exactly 1?
How many different irreducible monic polynomials of degree exactly 2?

How many different ways can you multiply these to get a polynomial of degree exactly 10?
 

1. What are irreducible polynomials?

Irreducible polynomials are polynomials that cannot be factored into smaller polynomials with coefficients in the same field. In other words, they cannot be broken down into simpler terms.

2. How can I determine if a polynomial is irreducible?

One way to determine if a polynomial is irreducible is to try to factor it using different methods, such as factoring by grouping or using the rational root theorem. If the polynomial cannot be factored, then it is irreducible.

3. What is the significance of irreducible polynomials?

Irreducible polynomials have important applications in algebra and number theory. They are often used in coding theory, cryptography, and other areas of mathematics.

4. Can irreducible polynomials have real coefficients?

Yes, irreducible polynomials can have real coefficients. They can have coefficients in any field, including real numbers, complex numbers, and finite fields.

5. How are irreducible polynomials related to prime numbers?

There is a connection between irreducible polynomials and prime numbers. In some cases, the roots of irreducible polynomials can be used to generate prime numbers, and the degree of the polynomial can determine the size of the prime numbers. However, this relationship is not always true and is still an active area of research.

Similar threads

I Looking for a creative or quick method for finding roots in GF(p^n)
    • Linear and Abstract Algebra
Replies
3
Views
739
I I think I've hit upon a very easy proof of impossibility of quintic
    • Linear and Abstract Algebra
Replies
2
Views
946
I What can be deduced about the roots of this polynomial?
    • Linear and Abstract Algebra
Replies
3
Views
1K
I Separable Polynomials - Paul E Bland's definition and exampl
    • Linear and Abstract Algebra
Replies
8
Views
2K
MHB Separable Polynomials - Paul E Bland's definition and example ....
    • Linear and Abstract Algebra
Replies
1
Views
1K
MHB Prove that there are infinitely many irreducible polynomials in Z5.
    • Linear and Abstract Algebra
Replies
3
Views
3K
MHB Splitting Fields and Separable Polynomials ....
    • Linear and Abstract Algebra
Replies
1
Views
1K
I Splitting Fields and Separable Polynomials ....
    • Linear and Abstract Algebra
Replies
6
Views
3K
MHB Understanding a problem in ring theory
    • Linear and Abstract Algebra
Replies
5
Views
2K
I Confused about Gauss's Lemma of Irreducibility
    • Linear and Abstract Algebra
Replies
4
Views
1K

Hot Threads

Recent Insights

Back
Top