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# 数学代写|代数数论代写Algebraic Number Theory代考|MAS6220 Preliminaries

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## 数学代写|代数数论代写Algebraic Number Theory代考|Preliminaries

We can discuss the concept of divisibility for any commutative ring $R$ with identity. Indeed, if $a, b \in R$, we will write $a \mid b(a$ divides $b)$ if there exists some $c \in R$ such that $a c=b$. Any divisor of 1 is called a unit. We will say that $a$ and $b$ are associates and write $a \sim b$ if there exists a unit $u \in R$ such that $a=b u$. It is easy to verify that $\sim$ is an equivalence relation.
Further, if $R$ is an integral domain and we have $a, b \neq 0$ with $a \mid b$ and $b \mid a$, then $a$ and $b$ must be associates, for then $\exists c, d \in R$ such that $a c=b$ and $b d=a$, which implies that $b d c=b$. Since we are in an integral domain, $d c=1$, and $d, c$ are units.

We will say that $a \in R$ is irreducible if for any factorization $a=b c$, one of $b$ or $c$ is a unit.

Example 2.1.1 Let $R$ be an integral domain. Suppose there is a map $n: R \rightarrow \mathbb{N}$ such that:
(i) $n(a b)=n(a) n(b) \forall a, b \in R$; and
(ii) $n(a)=1$ if and only if $a$ is a unit.
We call such a map a norm map, with $n(a)$ the norm of $a$. Show that every element of $R$ can be written as a product of irreducible elements.

Solution. Suppose $b$ is an element of $R$. We proceed by induction on the norm of $b$. If $b$ is irreducible, then we have nothing to prove, so assume that $b$ is an element of $R$ which is not irreducible. Then we can write $b=a c$ where neither $a$ nor $c$ is a unit. By condition (i),
$$n(b)=n(a c)=n(a) n(c)$$
and since $a, c$ are not units, then by condition (ii), $n(a)<n(b)$ and $n(c)<$ $n(b)$.

## 数学代写|代数数论代写Algebraic Number Theory代考|Gaussian Integers

Let $\mathbb{Z}[i]={a+b i \mid a, b \in \mathbb{Z}, i=\sqrt{-1}}$. This ring is often called the ring of Gaussian integers.
Exercise 2.2.1 Show that $\mathbb{Z}[i]$ is Euclidean.
Exercise 2.2.2 Prove that if $p$ is a positive prime, then there is an element $x \in \mathbb{F}_p:=\mathbb{Z} / p \mathbb{Z}$ such that $x^2 \equiv-1(\bmod p)$ if and only if either $p=2$ or $p \equiv 1$ $(\bmod 4)$. (Hint: Use Wilson’s theorem, Exercise 1.4.10.)
Exercise 2.2.3 Find all integer solutions to $y^2+1=x^3$ with $x, y \neq 0$.
Exercise 2.2.4 If $\pi$ is an element of $R$ such that when $\pi \mid a b$ with $a, b \in R$, then $\pi \mid a$ or $\pi \mid b$, then we say that $\pi$ is prime. What are the primes of $\mathbb{Z}[i]$ ?

Exercise 2.2.5 A positive integer $a$ is the sum of two squares if and only if $a=b^2 c$ where $c$ is not divisible by any positive prime $p \equiv 3(\bmod 4)$.

## 数学代写|代数数论代写Algebraic Number Theory代考|Preliminaries

ii) $n(a)=1$ 当且仅当 $a$ 是一个单位。

$$n(b)=n(a c)=n(a) n(c)$$

## MATLAB代写

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