How can a prime element in a Ring be proven as irreducible?

In summary, Number Theory is a branch of mathematics that deals with the properties and relationships of integers and their patterns. It has various applications in cryptography, coding theory, and computer science, and helps in analyzing and predicting patterns in data. Some well-known unsolved problems in Number Theory include the Goldbach Conjecture, the Riemann Hypothesis, and the Collatz Conjecture. It is closely connected to algebra, geometry, and analysis and can be challenging to learn, but with dedication and practice, it can be mastered. Overall, it is a fascinating subject with many real-world applications.
  • #1
NoodleDurh
23
0
How does one show that a prime element in a Ring is irreducible and how does one show that ##|| x || = 1## iff x is a unit.

okay from my knowledge I know that units are invertible elements, so how does the norm of x make it 1... maybe I am not too sure about this
 
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  • #2
For your first question, try doing a proof by contradiction. Let p be a prime element, and suppose it is reducible. What can you do with that?
 

1) What is Number Theory?

Number Theory is a branch of mathematics that deals with the properties and relationships of integers and their patterns.

2) How is Number Theory used in real life?

Number Theory has various applications in cryptography, coding theory, and computer science. It also helps in analyzing and predicting patterns in data.

3) What are some famous unsolved problems in Number Theory?

Some well-known unsolved problems in Number Theory include the Goldbach Conjecture, the Riemann Hypothesis, and the Collatz Conjecture.

4) How is Number Theory related to other branches of mathematics?

Number Theory is closely connected to algebra, geometry, and analysis. Many concepts in these fields, such as prime numbers and divisibility, are studied in Number Theory as well.

5) Is Number Theory a difficult subject to learn?

Like any other branch of mathematics, Number Theory can be challenging, but with dedication and practice, it can be mastered. It is also a fascinating subject with many real-world applications.

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