Algebra question (irreducibility of polynomial)

Thread starter nicolep
Start date Nov 25, 2007
Tags

Algebra Polynomial

In summary, the irreducibility of a polynomial is a property that determines whether a polynomial can be factored into smaller polynomials with lower degrees. To determine if a polynomial is irreducible, one can use the rational root theorem, the Eisenstein's criterion, or the GCD test. A polynomial with only real coefficients can be irreducible, and it is possible for a polynomial to be irreducible over one set of numbers but reducible over another set. The significance of knowing if a polynomial is irreducible lies in its practical applications in fields such as cryptography, coding theory, and number theory, as well as its role in understanding the structure and behavior of polynomials.

Nov 25, 2007

nicolep

Homework Statement

Show that this polynomial is irreducible over Q[x]

Homework Equations

x^4 + 6x^3 + 14x^2 + 16x + 9

The Attempt at a Solution

I think Eisenstein's criterion will have to be used. But is some substitution to be made??
Please help!
Thanks

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Nov 25, 2007

morphism

Science Advisor

Homework Helper

Try x=y+1.

1. What is the definition of irreducibility of a polynomial?

The irreducibility of a polynomial is a property that determines whether a polynomial can be factored into smaller polynomials with lower degrees.

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

To determine if a polynomial is irreducible, you can use the rational root theorem, the Eisenstein's criterion, or the GCD test. These methods involve checking for the existence of rational roots, prime factors, or common factors respectively.

3. Can a polynomial with only real coefficients be irreducible?

Yes, a polynomial with only real coefficients can be irreducible. For example, the polynomial x^2 + 1 is irreducible over the real numbers.

4. Is it possible for a polynomial to be irreducible over one set of numbers but reducible over another set?

Yes, it is possible for a polynomial to be irreducible over one set of numbers but reducible over another set. For instance, the polynomial x^2 + 1 is irreducible over the integers but reducible over the complex numbers.

5. What is the significance of knowing if a polynomial is irreducible?

The irreducibility of a polynomial has practical applications in fields such as cryptography, coding theory, and number theory. It also helps in understanding the structure and behavior of polynomials, which are important in many areas of mathematics and science.