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# 数学代写|数论代写Number Theory代考|MATH2301 The Finite Fields Fpn

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## 数学代写|数论代写Number Theory代考|The Finite Fields Fpn

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|数论代写Number Theory代考|The Order of a Finite Field

From our previous work we know exactly the structure of the finite field $\mathbb{F}_p$ where $p$ is prime. Specifically, this field has $p$ elements denoted ${0,1, \ldots, p-1}$ and its operations of addition and multiplication are carried out modulo $p$. However, if $F$ is a finite field but is not one of these, we know little about it except that its order (i.e, its number of elements) is finite. Theorem 10.1, which we stated without proof (as yet), says that this order is a prime power, but we need to prove this fact. Moreover, we know little about the algebraic structure of $F$; that is, how do we perform addition and multiplication? In this section we will concentrate on the former question about order, and then in Section $10.4$ we’ll turn to the question about structure.

In order to explore these matters, we need to be reminded of (or learn) some basic ideas from linear and abstract algebra. Our intent here is not to provide every detail but only to provide the necessary basics.
(1) If $F$ is any field and $K$ is a subset of $F$, then $K$ is a subfield of $F$ if $K$ is itself a field under the operations of $F$. So, for example, the set $\mathbb{Q}$ of rational numbers is a subfield of the set $\mathbb{R}$ of real numbers since the sum, difference, product, and quotient of two rational numbers is a rational number. We note in passing that the set $\mathbb{Z}$ of integers is not a subfield of $\mathbb{R}$ since $\mathbb{Z}$ is not itself a field (only 1 and $-1$ have multiplicative inverses in $\mathbb{Z}$ ).

(2) If $K$ is any field, a vector space $V$ over $K$ is a set which has an addition operation (in the language of abstract algebra, it is an “additive Abelian group”), and it also has a “scalar multiplication” by elements of $K$; that is, if $v$ is in $V$ and $\lambda$ is in $K$, then the product $\lambda v$ must also be in $V$. If there is a set $B=\left{b_1, \ldots, b_n\right}$ of elements of $V$ such that every $v \in V$ has a unique representation $v=\lambda_1 b_1+\cdots+\lambda_n b_n$ (where each $\lambda_i$ is in $K$ ), then we say that $B$ is a basis for $V$ over $K$ and we say that the dimension of $V$ over $K$ is $n$.
(3) The characteristic of a field $F$ is the smallest number $k$ such that
$$\underbrace{1+1+\cdots+1}{k \text { terms }}=0 .$$ If no such $k$ exists, we say that the characteristic is 0 ; for example, the set of real numbers $\mathbb{R}$ is an infinite field of characteristic 0 . Also note that $k$, if it exists, must be prime, for if $k=a \cdot b$ with $a, b>1$, then $$0=\underbrace{1+1+\cdots+1}{k \text { terms }}=(\underbrace{1+1+\cdots+1}{a \text { terms }})(\underbrace{1+1+\cdots+1}{b \text { terms }}),$$

so one of these factors must be 0 , contradicting the minimality of $k$.

We can now use the concepts of subfield and vector space to pin down the possible order of a finite field, as stated in Theorem 10.1. Below, the notation $|F|$ means the order of $F$; i.e., the number of elements in $F$.

## 数学代写数论代写Number Theory代考|The Order of a Finite Field

(1) 如果 $F$ 是任何领域并且 $K$ 是的一个子集 $F$ ，然后 $K$ 是一个子域 $F$ 如果 $K$ 本身就是 个领域 $F$. 因此，例如，集合 $\mathbb{Q}$ 有理数是集 段 (只有1和 $-1$ 有乘法逆元 $\mathbb{Z})$.
（2）如果 $K$ 是任意场，向量空间 $V$ 超过 $K$ 是一个具有加法枟算的集合（在抽象代数的语言中，它是一个“加法阿贝尔群”)，并且它 还具有与元嫊的“标量乘法” $K$; 也就是脱，如果 $v$ 在 $V$ 和 $\lambda$ 在 $K$ ，那 $\angle$ 产品 $\lambda v$ 也必须在 $V$. 如果有一套
\eft 的分隔符缺失或无法识别 的元筙 $V$ 这样每一个 $v \in V$ 具有独恃的代表性 $v=\lambda_1 b_1+\cdots+\lambda_n b_n$ (其中 每个 $\lambda_i$ 在 $\left.K\right)$ ，那么我们说 $B$ 是一个基础 $V$ 超过 $K$ 我们说的维度 $V$ 超过 $K$ 是 $n$.
(3) 领域的特点 $F$ 是最小的数 $k$ 这样
$$\underbrace{1+1+\cdots+1} k \text { terms }=0 .$$

$$0=\underbrace{1+1+\cdots+1} k \text { terms }=(\underbrace{1+1+\cdots+1} a \text { terms })(\underbrace{1+1+\cdots+1} b \text { terms }),$$

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