# Is $2 + 8\sqrt{-5}$ a Unit or Irreducible in $\mathbb{Z} + \mathbb{Z}\sqrt{-5}$?

• MHB
• abs1
In summary: Yes, it does.Next, we consider $8$.Is it a unit?No, it is not.Why not?Because it has a non-zero inverse, which is not a unit in the given ring.
abs1
prove that $2+8{\sqrt{-5}}$ is unit and irreducible or not in $\mathbb Z+\mathbb Z{\sqrt{-5}}$.

Last edited by a moderator:
abs said:
prove that $2+8{\sqrt{-5}}$ is unit and irreducible or not in $\mathbb Z+\mathbb Z{\sqrt{-5}}$.

Hint: we can write $2+8{\sqrt{-5}}=2(1+4\sqrt{-5})$.

Klaas van Aarsen said:
Hint: we can write $2+8{\sqrt{-5}}=2(1+4\sqrt{-5})$.

please explain in detail if possible

abs said:
please explain in detail if possible

What is the definition of a unit?

an element alpha belong to k ia called a unit if alpha divisible by 1.
dear it is my question if u not solved it then no problem its ok .if u solved it then give me complete explanation thank u so much
irreducible element:a non zero non unit element alpha belong to k is said to be irreducible if aplha=ab.
either a is unit or b is unit.
i give u both def. of unit and irreducible thank u so much

Last edited by a moderator:
abs said:
an element alpha belong to k ia called a unit if alpha divisible by 1.

Not quite.
From wiki:

a unit in a ring with identity $R$ is any element $u$ that has an inverse element in the multiplicative monoid of $R$, i.e. an element $v$ such that
$$uv = vu = 1_R,$$
where $1_R$ is the multiplicative identity​

abs said:
dear it is my question if u not solved it then no problem its ok .if u solved it then give me complete explanation thank u so much

Sorry, we are a math help site.
We do not usually give complete solutions.
Instead we give hints or similar to help people to learn math.

abs said:
irreducible element:a non zero non unit element alpha belong to k is said to be irreducible if aplha=ab.
either a is unit or b is unit.

If you're up to it...

The hint I gave showed that we can split the expression in two factors that we might call $a$ and $b$.
Let's start with $2$.
Is it a unit? That is, does it have a multiplicative inverse in the given ring?

## 1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations, vectors, matrices, and their properties. It is used to solve problems involving systems of linear equations and to analyze geometric transformations.

## 2. Why is linear algebra important?

Linear algebra is a fundamental tool in many fields of science and engineering, such as physics, computer science, economics, and statistics. It provides a powerful framework for solving complex problems and understanding the relationships between variables.

## 3. What are the basic concepts in linear algebra?

The basic concepts in linear algebra include vectors, matrices, systems of linear equations, vector spaces, linear transformations, eigenvalues and eigenvectors, and determinants. These concepts are used to represent and solve problems in a variety of applications.

## 4. How is linear algebra used in data analysis?

Linear algebra is essential for data analysis as it provides tools for organizing, manipulating, and analyzing large datasets. It is used in data mining, machine learning, and statistical analysis to identify patterns, make predictions, and draw conclusions from data.

## 5. What are some real-world applications of linear algebra?

Linear algebra has a wide range of real-world applications, including image and signal processing, computer graphics, robotics, cryptography, and optimization. It is also used in finance to model and analyze financial data and in engineering to design and control systems.

• Linear and Abstract Algebra
Replies
6
Views
1K
• Linear and Abstract Algebra
Replies
4
Views
1K
• Linear and Abstract Algebra
Replies
11
Views
1K
• Calculus and Beyond Homework Help
Replies
15
Views
2K
• Linear and Abstract Algebra
Replies
14
Views
2K
• Linear and Abstract Algebra
Replies
6
Views
2K
• Linear and Abstract Algebra
Replies
8
Views
1K
• Linear and Abstract Algebra
Replies
6
Views
1K
• Precalculus Mathematics Homework Help
Replies
10
Views
491
• Linear and Abstract Algebra
Replies
6
Views
2K