Proving R[x] is a Principal Ideal Domain Implies R is a Field

In summary, the problem is asking to show that if R is an integral domain and R[x] is a principal ideal domain, then R is a field. This means that every non-zero element in R has a multiplicative inverse, which is the main difference between a field and an integral domain. It is recommended to start by understanding the definitions of "field" and "integral domain" before attempting the problem.
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Homework Statement



Let R be an integral domain and suppose that R[x] is a principal ideal domain. Show that R is a field.


Homework Equations




I don't know where to start, I'm not familiar with this material. I was browsing through an abstract algebra book and found this. Would like an explanation of what it means, thank you.

The Attempt at a Solution

 
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The difference between an integral domain and a field is that every non-zero member of a field has a multiplicative inverse. That is what you need to prove.

(If you are not familiar with the material, why are you trying to do this problem? Wouldn't it be better to start at the beginning and read the book rather than "browsing"? Do you need definitions of "field" and "integral domain"?)
 

FAQ: Proving R[x] is a Principal Ideal Domain Implies R is a Field

What is Abstract Algebra Proof?

Abstract Algebra Proof is a mathematical method used to prove theorems and propositions in abstract algebra, a branch of mathematics that studies algebraic structures such as groups, rings, and fields. It involves using logic and mathematical reasoning to demonstrate the validity of a statement or claim.

Why is Abstract Algebra Proof important?

Abstract Algebra Proof is important because it allows us to formally prove mathematical statements, which helps us gain a deeper understanding of the concepts and structures in abstract algebra. It also provides a solid foundation for further research and applications in the field.

What are the key elements of an Abstract Algebra Proof?

The key elements of an Abstract Algebra Proof include defining the problem, stating the assumptions and axioms, presenting the steps in the proof, and providing a logical and rigorous argument to support the conclusion. It also involves the use of definitions, theorems, and previously proven results.

What are some common strategies used in Abstract Algebra Proof?

Some common strategies used in Abstract Algebra Proof include using direct proof, proof by contradiction, mathematical induction, and proof by cases. It is also important to understand the properties and characteristics of the algebraic structures being studied in order to develop an effective proof.

How can one improve their skills in Abstract Algebra Proof?

To improve skills in Abstract Algebra Proof, one can practice by working on a variety of problems and proofs, familiarize themselves with common definitions and theorems, and seek guidance from experts in the field. It is also helpful to read and analyze published proofs to gain a better understanding of the techniques used.

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