Example Of Nonconstant Polynomial Over A Finite Field With Zero Derivative

Introduction

In abstract algebra, polynomials over finite fields have unique properties that distinguish them from polynomials over fields of characteristic zero. One of the most interesting properties is that a nonconstant polynomial over a finite field can have a zero derivative. In this article, we will explore this concept and provide an example of a nonconstant polynomial over a finite field with zero derivative.

Background

To understand the concept of a nonconstant polynomial over a finite field with zero derivative, we need to recall some basic definitions and properties.

• A field is a set of elements with two binary operations, addition and multiplication, that satisfy certain properties.
• A finite field is a field with a finite number of elements.
• A polynomial over a field is an expression of the form $f(t) = \sum_{k = 0}^n a_k t^k$, where $a_k$ are elements of the field and $t$ is an indeterminate.
• The derivative of a polynomial $f(t)$ is the polynomial $Df(t) = \sum_{k = 1}^n ka_k t^{k-1}$.

Properties of Polynomials Over Finite Fields

Polynomials over finite fields have some unique properties that are not shared by polynomials over fields of characteristic zero. One of the most interesting properties is that a nonconstant polynomial over a finite field can have a zero derivative.

Theorem 1

If $K$ is a field of characteristic zero, then a polynomial $f(t) = \sum_{k = 0}^n a_k t^k \in K[t]$ with zero derivative $Df(t) = 0$ must be constant.

Proof

Let $f(t) = \sum_{k = 0}^n a_k t^k$ be a polynomial over a field of characteristic zero with zero derivative $Df(t) = 0$. Then, we have:

$$Df(t) = \sum_{k = 1}^n ka_k t^{k-1} = 0$$

Since the characteristic of the field is zero, we can multiply both sides of the equation by $t$ without changing the equation. This gives us:

$$\sum_{k = 1}^n ka_k t^k = 0$$

Now, we can use the fact that the field has characteristic zero to conclude that $a_k = 0$ for all $k \geq 1$. This is because, in a field of characteristic zero, the equation $ka_k = 0$ implies that $a_k = 0$.

Therefore, we have $f(t) = a_0$, which is a constant polynomial.

Counterexample

However, this result does not hold for fields of characteristic $p > 0$. In fact, we can construct a nonconstant polynomial over a finite field with zero derivative.

Example

Let $K = \mathbb{F}_p$ be a finite field of characteristic $p > 0$. Consider the polynomial:

$$f(t) = t^p - t$$

This polynomial is nonconstant, but its derivative is zero:

Df(t) = pt^{p-1 - 1 = 0

This is because $p$ is a prime number, and $pt^{p-1}$ is always zero in a field of characteristic $p$.

Conclusion

In conclusion, we have shown that a nonconstant polynomial over a finite field can have a zero derivative. This is in contrast to the result for fields of characteristic zero, where a polynomial with zero derivative must be constant. The example we provided demonstrates that this result does not hold for fields of characteristic $p > 0$.

Further Research

This result has implications for the study of polynomials over finite fields. Further research is needed to fully understand the properties of polynomials over finite fields and to explore the consequences of this result.

References

• [1] Lang, S. (1993). Algebra. Springer-Verlag.
• [2] Artin, E. (1947). Galois Theory. Notre Dame Mathematical Lectures, 2.
• [3] van der Waerden, B. L. (1930). Moderne Algebra. Springer-Verlag.

Glossary

• Field: A set of elements with two binary operations, addition and multiplication, that satisfy certain properties.
• Finite field: A field with a finite number of elements.
• Polynomial: An expression of the form $f(t) = \sum_{k = 0}^n a_k t^k$, where $a_k$ are elements of the field and $t$ is an indeterminate.
• Derivative: The polynomial $Df(t) = \sum_{k = 1}^n ka_k t^{k-1}$.
• Characteristic: The smallest positive integer $p$ such that $p \cdot 1 = 0$ in a field.
  Q&A: Nonconstant Polynomial Over a Finite Field with Zero Derivative ====================================================================

Q: What is a nonconstant polynomial over a finite field?

A: A nonconstant polynomial over a finite field is a polynomial that cannot be expressed as a constant multiple of a polynomial of lower degree. In other words, it is a polynomial that has at least one term with a non-zero coefficient.

Q: What is a finite field?

A: A finite field is a field with a finite number of elements. In other words, it is a set of elements with two binary operations, addition and multiplication, that satisfy certain properties, and the set has a finite number of elements.

Q: What is the characteristic of a field?

A: The characteristic of a field is the smallest positive integer $p$ such that $p \cdot 1 = 0$ in the field. In other words, it is the smallest positive integer $p$ such that $p$ times the multiplicative identity of the field is equal to the additive identity of the field.

Q: What is the derivative of a polynomial?

A: The derivative of a polynomial is a polynomial that is obtained by differentiating the original polynomial. It is denoted by $Df(t)$ and is defined as the polynomial $Df(t) = \sum_{k = 1}^n ka_k t^{k-1}$.

Q: Can a nonconstant polynomial over a finite field have a zero derivative?

A: Yes, a nonconstant polynomial over a finite field can have a zero derivative. This is in contrast to the result for fields of characteristic zero, where a polynomial with zero derivative must be constant.

Q: What is an example of a nonconstant polynomial over a finite field with zero derivative?

A: An example of a nonconstant polynomial over a finite field with zero derivative is the polynomial $f(t) = t^p - t$, where $p$ is a prime number and $t$ is an indeterminate.

Q: Why does the derivative of $f(t) = t^p - t$ equal zero?

A: The derivative of $f(t) = t^p - t$ equals zero because $p$ is a prime number, and $pt^{p-1}$ is always zero in a field of characteristic $p$.

Q: What are the implications of this result?

A: This result has implications for the study of polynomials over finite fields. Further research is needed to fully understand the properties of polynomials over finite fields and to explore the consequences of this result.

Q: What are some open questions in this area of research?

A: Some open questions in this area of research include:

• What are the necessary and sufficient conditions for a nonconstant polynomial over a finite field to have a zero derivative?
• Can we generalize this result to polynomials over other types of rings?
• What are the implications of this result for the study of algebraic geometry over finite fields?

Q: How can I learn more about this topic?

A: To learn more about this topic, you can start by reading the references listed at the end of this article. You can also try searching for online resources, such as lecture notes or research papers, that discuss this topic. Additionally, you can try contacting a mathematician who specializes in this area of research for more information.

References

• [1] Lang, S. (1993). Algebra. Springer-Verlag.
• [2] Artin, E. (1947). Galois Theory. Notre Dame Mathematical Lectures, 2.
• [3] van der Waerden, B. L. (1930). Moderne Algebra. Springer-Verlag.

Glossary

• Field: A set of elements with two binary operations, addition and multiplication, that satisfy certain properties.
• Finite field: A field with a finite number of elements.
• Polynomial: An expression of the form $f(t) = \sum_{k = 0}^n a_k t^k$, where $a_k$ are elements of the field and $t$ is an indeterminate.
• Derivative: The polynomial $Df(t) = \sum_{k = 1}^n ka_k t^{k-1}$.
• Characteristic: The smallest positive integer $p$ such that $p \cdot 1 = 0$ in a field.