Is Every Rootless Polynomial Over a Finite Field Prime?

In summary, a polynomial of degree 2 or 3 over a field F is a prime polynomial if and only if it does not have a root in F. However, it is possible for a polynomial of degree 4 over a field F to have no root in F and still not be a prime polynomial. This is because each polynomial f(x) in F[x] determines a polynomial function from F to F, and it can be proven that different polynomials determine different functions when F is a finite field.
  • #1
wxrebecca
6
0
How to prove that a polynomial of degree 2 or 3 over a filed F is a prime polynomial if and only if the polynomial does not have a root in F?

and i can't think of an example of polynomial of degree 4 over a field F that has no root in F but is not a prime polynomial.

it says each polynomial f(x) in F[x] determines a function from F to F by the rule c--> f(c). such a function is called a polynomial function from F to F. how to prove the different polynomials determine different functions when F is an finite field?


thanks for the help

Rebecca
 
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  • #2
the most basic result on roots of polynomials says that r is a root if and only if (x-r) is a factor.
 

FAQ: Is Every Rootless Polynomial Over a Finite Field Prime?

What is a polynomial?

A polynomial is a mathematical expression that consists of variables, coefficients, and arithmetic operations such as addition, subtraction, multiplication, and non-negative integer exponents.

What are the different types of polynomials?

The three main types of polynomials are monomials, binomials, and trinomials. A monomial is a polynomial with one term, a binomial has two terms, and a trinomial has three terms.

How do you factor a polynomial?

To factor a polynomial, you need to find its greatest common factor (GCF) and use the distributive property to write the polynomial as a product of its GCF and another polynomial. Then, you can continue factoring the remaining polynomial until it is in its simplest form.

How do you solve polynomial equations?

To solve a polynomial equation, you can use various methods such as factoring, the quadratic formula, or the rational root theorem. Factoring involves finding the roots of the polynomial, while the quadratic formula and the rational root theorem use specific formulas to find the roots.

What are some real-life applications of polynomials?

Polynomials have many applications in different fields such as physics, economics, and engineering. They are used to model real-world phenomena, make predictions, and solve problems. For example, polynomials can be used to calculate the trajectory of a projectile, determine the optimal price for a product, or design a bridge that can withstand certain forces.

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