Exploring the Roots of Irreducible Polynomials of Small Degree

  • Thread starter letmeknow
  • Start date
  • Tags
    Degree
In summary, the proposition states that a polynomial of degree 2 or 3 over a field F is reducible if and only if it has a root in F. This was demonstrated by the example of x^4 + 3x^2 + 2 which is reducible but has no roots in Q, serving as a counterexample for polynomials of degree 4. This is why the proposition only applies to polynomials of degree 2 or 3. Additionally, for a polynomial of degree 3 to be reducible, it must be reduced to either two linear factors or a linear factor and an irreducible quadratic, both of which will result in at least one root.
  • #1
letmeknow
26
0
Proposition. A polynomial of degree 2 or 3 over a field F is reducible iff it has a root in F.

Tell me if I'm on the right track... I see that x^4 + 3x^2 + 2 is reducible (x^2+1)(x^2+2) but has no roots in Q.

This serves as a counterexample to the proposition if states for polynomial up to degree 4?
 
Physics news on Phys.org
  • #2
Yes, that serves as a counter example for polynomials of degree 4. And that's why your proposition only says "degree 2 or 3".

If a polynomial of degree three is "reducible", then it must be "reduced" to two linear factors. And those two linear factors will give roots. If a polynomial of degree three is "reducible" then it is "reduced" to either a product of three linear factors or a product of a linear factor and an irreducible quadratic. In either case, you have at least one linear factor that gives a root.
 
Last edited by a moderator:

What are irreducibles of small degree?

Irreducibles of small degree refer to polynomials with the smallest possible degree that cannot be factored into smaller polynomials with integer coefficients. These polynomials are also known as prime polynomials.

Why are irreducibles of small degree important?

Irreducibles of small degree play a crucial role in algebraic number theory and algebraic geometry. They are used to study the structure of number fields and algebraic curves, and are fundamental in the study of prime factorization of integers.

How can one determine if a polynomial is irreducible of small degree?

To determine if a polynomial is irreducible of small degree, one can use various methods such as the Eisenstein criterion, the rational root test, and the Berlekamp algorithm. These methods help in determining if a polynomial is irreducible or can be factored into smaller polynomials with integer coefficients.

What are some examples of irreducibles of small degree?

Some examples of irreducibles of small degree include the polynomial x^2 + 1, which is irreducible over the real numbers but can be factored into (x+i)(x-i) over the complex numbers. Another example is the polynomial x^3 - 2, which is irreducible over the rational numbers but can be factored into (x-2)(x^2 + 2x + 4) over the real numbers.

How are irreducibles of small degree related to prime numbers?

Irreducibles of small degree are closely related to prime numbers in number theory. In fact, every prime number can be expressed as an irreducible polynomial of degree 1. This connection is also useful in understanding the distribution of prime numbers and their relationship to polynomials.

Similar threads

I Looking for a creative or quick method for finding roots in GF(p^n)
    • Linear and Abstract Algebra
Replies
3
Views
748
MHB The polynomial is irreducible over Q(i)
    • Linear and Abstract Algebra
Replies
16
Views
2K
I I think I've hit upon a very easy proof of impossibility of quintic
    • Linear and Abstract Algebra
Replies
2
Views
956
I Finite fields, irreducible polynomial and minimal polynomial theorem
    • Linear and Abstract Algebra
Replies
6
Views
2K
MHB Proof: K is a Root Field for Every Irreducible Polynomial with a Root in K
    • Linear and Abstract Algebra
Replies
1
Views
1K
I Separable Polynomials - Paul E Bland's definition and exampl
    • Linear and Abstract Algebra
Replies
8
Views
2K
MHB Question on the Irreducibility of Polynomials
    • Linear and Abstract Algebra
Replies
1
Views
1K
MHB Separable Polynomials - Paul E Bland's definition and example ....
    • Linear and Abstract Algebra
Replies
1
Views
1K
I Generators of Galois group of ## X^n - \theta ##
    • Linear and Abstract Algebra
Replies
3
Views
1K
MHB Eisenstein polynomial and field extension
    • Linear and Abstract Algebra
Replies
24
Views
4K

Hot Threads

Recent Insights

Back
Top