# 数学代写|数论作业代写number theory代考|MATH2088

## 数学代写|数论作业代写number theory代考|The Finite Fields

To emphasize that we have a field, from here on we shall denote the finite field with $p$ elements by $\mathbb{F}_p$ rather than $\mathbb{Z}_p$. This is simply a notational change; $\mathbb{F}_p$ and $\mathbb{Z}_p$ both denote the exact same set with the exact same algebraic structure.

By a prime power we mean a positive integer of the form $p^e$ where $p$ is a prime and $e \geq 1$ is a positive integer. For example, $2^4$, $5^{10}$, and $97^2$ are prime powers, but numbers like 6, 10, 12, 24, 96 are not prime powers. We note that prime numbers are also prime powers since the exponent $e$ can be 1 . Below we shall often use $q$ to represent a prime power; that is, $q=p^e$ for some prime $p$ and some positive integer exponent $e$.

Here then is the answer to our question above about the existence of finite fields besides the collection $\left{\mathbb{F}_p \mid p\right.$ prime $}$.

Theorem 10.1. (a) If $p$ is a prime and $e \geq 1$ is a positive integer, there is a finite field $\mathbb{F}_{p^e}$ of order $p^e$, i.e., which contains exactly $p^e$ distinct elements.
(b) Moreover if $F$ is a finite field, then $F$ must contain exactly $p^e$ distinct elements for some prime $p$ and some positive integer $e \geq 1$

## 数学代写|数论作业代写number theory代考|Constructing Finite Fields

We now turn to the following question: Given a prime number $p$ and a positive integer $e$, how can we construct both the elements and the arithmetic operations of a finite field $\mathbb{F}_{p^e}$ ? Of course we’ve long since known how to do this when $e=1$, but we have to this point not considered how to do it when $e>1$. It turns out that a convenient way to do this construction is to utilize polynomials in a single unknown whose coefficients are taken from the prime finite field $\mathbb{F}_p$. We shall denote by $\mathbb{F}_p[\theta]$ the set of all polynomials in the single unknown $\theta$ with coefficients in $\mathbb{F}_p$, where $p$ is any prime. This set has algebraic structure by using standard addition and multiplication of polynomials. Here are two important definitions:
A polynomial $f(\theta)$ is called monic if its leading term (i.e., nonzero term of highest degree) has a coefficient of 1 . So, for example $\theta^2+4 \theta+3$ is monic but $2 \theta+4$ is not. Important idea: Monic polynomials in $\mathbb{F}_p[\theta]$ are the analogue of positive numbers in the integers $\mathbb{Z}$.

A polynomial $f(\theta)$ is called irreducible if it cannot be factored into two polynomials of positive degree. For example, in $\mathbb{F}_2[\theta], \theta^2+\theta+1$ is irreducible, but $\theta^2+1=(\theta+1)(\theta+1)$ is not. We note that the irreducibility of a polynomial depends on the field of coefficients. For example, the polynomial $x^2-2$ is irreducible over the field of $\mathbb{Q}$ of rational numbers, but it is reducible over the field $\mathbb{R}$ of real numbers since there $x^2-2=(x-\sqrt{2})(x+\sqrt{2})$. Important idea: Irreducible polynomials in $\mathbb{F}_p[\theta]$ are the analogue of prime numbers in the integers $\mathbb{Z}$.

# 数论代考

## 数学代写|数论作业代写数论代考|有限域

(b)此外，如果 $F$ 是有限域吗 $F$ 必须包含精确的 $p^e$ 质数的不同元素 $p$ 和某个正整数 $e \geq 1$

## 数学代写|数论作业代写数论代考|构造有限域

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