# Gaussian integers, ring homomorphism and kernel

• rayman123
In summary, the conversation discusses a map \varphi from \mathbb{Z}[i] to \mathbb{Z}_{2}, and its properties as a ring homomorphism. The kernel of \varphi is found to be the set of Gaussian integers a+bi that satisfy ker\varphi=(a+bi). It is also shown that ker\varphi is a maximal ideal in \mathbb{Z}[i]. However, there is a discrepancy in the multiplication operation that may be due to bd being congruent to -bd (mod 2). Further assistance is sought in finding the kernel of \varphi.
rayman123

## Homework Statement

let $$\varphi:\mathbb{Z}\rightarrow \mathbb{Z}_{2}$$ be the map for which $$\varphi(a+bi)=[a+b]_{2}$$
a)verify that $$\varphi$$ is a ring homomorphism and determine its kernel
b) find a Gaussian integer z=a+bi s.t $$ker\varphi=(a+bi)$$
c)show that $$ker\varphi$$ is maximal ideal in $$\mathbb{Z}$$

I started by showing that $$\varphi$$ preserves the ring operations
$$\varphi((a+bi)+(c+di))=\varphi((a+c)+(b+d)i)=[(a+c)+(b+d)]_{2}=[a+b]_{2}\oplus[c+d]_{2}=\varphi(a+bi)+\varphi(c+d)$$
and multiplication
$$\varphi((a+bi)(c+di))=\varphi(ac+adi+bic-bd)=\varphi((ac-bd)+(ad+bc)i)=[(ac-bd)+(ad+bc)]_{2}=ac-bd+ad+bc$$
but something is not right here because if I look at the right hand side, I should get
$$\varphi(a+bi)\varphi(c+di)=[a+b]_{2}[c+d]_{2}=[(a+b)(c+d)]_{2}=ac+ad+bc+bd$$...

I don't know how to find the kernel, I know that by def $$ker\varphi=\{z\in\mathbb{Z}; \varphi(z)=[0]_{2}\}$$

rayman123 said:

## Homework Statement

let $$\varphi:\mathbb{Z}\rightarrow \mathbb{Z}_{2}$$ be the map for which $$\varphi(a+bi)=[a+b]_{2}$$
a)verify that $$\varphi$$ is a ring homomorphism and determine its kernel
b) find a Gaussian integer z=a+bi s.t $$ker\varphi=(a+bi)$$
c)show that $$ker\varphi$$ is maximal ideal in $$\mathbb{Z}$$

I started by showing that $$\varphi$$ preserves the ring operations
$$\varphi((a+bi)+(c+di))=\varphi((a+c)+(b+d)i)=[(a+c)+(b+d)]_{2}=[a+b]_{2}\oplus[c+d]_{2}=\varphi(a+bi)+\varphi(c+d)$$
and multiplication
$$\varphi((a+bi)(c+di))=\varphi(ac+adi+bic-bd)=\varphi((ac-bd)+(ad+bc)i)=[(ac-bd)+(ad+bc)]_{2}=ac-bd+ad+bc$$
but something is not right here because if I look at the right hand side, I should get
$$\varphi(a+bi)\varphi(c+di)=[a+b]_{2}[c+d]_{2}=[(a+b)(c+d)]_{2}=ac+ad+bc+bd$$...

Maybe bd=-bd (mod 2) ??

I don't know how to find the kernel, I know that by def $$ker\varphi=\{z\in\mathbb{Z}; \varphi(z)=[0]_{2}\}$$

Yes, so take $a+bi \in \ker(\varphi$. Then $\varphi(a+bi)=0$. Now just write things out using the definition of $\varphi$.

## What are Gaussian integers?

Gaussian integers are a subset of complex numbers, also known as complex integers. They are numbers of the form a + bi, where a and b are integers and i is the imaginary unit. For example, 3 + 2i and -5 - 4i are Gaussian integers.

## What is a ring homomorphism?

A ring homomorphism is a function that preserves the structure of a ring. In other words, it is a function between two rings that preserves addition, multiplication, and identity elements. It also preserves the distributive property. An example of a ring homomorphism is the function f: Z -> Z, where f(n) = 2n.

## What is the kernel of a ring homomorphism?

The kernel of a ring homomorphism is the set of elements in the domain that map to the identity element in the codomain. In other words, it is the set of elements that are mapped to 0 by the homomorphism. The kernel of a ring homomorphism is always an ideal in the domain.

## How are Gaussian integers related to ring homomorphism?

Gaussian integers can be viewed as a ring, with addition and multiplication defined in the same way as complex numbers. A ring homomorphism can be defined between two rings, one of which is the ring of Gaussian integers. This allows for the study of the properties of Gaussian integers using ring homomorphisms.

## What is the significance of the kernel in ring homomorphism?

The kernel of a ring homomorphism is an important concept in abstract algebra, as it helps to understand the structure and properties of rings. It is also used in the study of quotient rings, which are rings obtained by dividing a ring by an ideal. The kernel is also used to define the concept of isomorphism, which is an important tool in understanding the relationship between different rings.

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