Is S Isomorphic to Any Commonly Known Fields?

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In summary, the ring S defined as the set of all pairs (q,s) with q and s being rational numbers, along with the operations of addition and multiplication defined as (q,s) + (q',s') = (q+q',s+s') and (q,s)(q',s') = (qs'+q's, 3qq'+ss'), respectively, has been proven to be a field. This was done by showing that S is isomorphic to the field of numbers of the form p+q*sqrt(3), where p and q are rational, which has dimension two as a vector space over the rationals.
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bokasabi
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Homework Statement


let S be the ring of all pairs (q,s) with q,s being rational.
define (q,s) + (q',s') = (q+q',s+s')
and (q,s)(q',s') = (qs'+q's,3qq'+ss')
Prove that S is a field.

Homework Equations




Try proving that it is isomorphic to something else using the First homomorphism theorem for rings.



The Attempt at a Solution



I can not find a homomorphism between S and any commonly known fields like complex numbers, real numbers, rational numbers, etc
 
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  • #2
How about commonly known extension fields? Like the rationals extended by sqrt(3)?
 
  • #3
I am not sure what that means
 
  • #4
The set of numbers of the form p+q*sqrt(3) where p and q are rational. It's a field. Can you prove it? It has dimension two as a vector space over the rationals. I think of that as pretty commonly known.
 
Last edited:
  • #5
got it, thanks
 

Related to Is S Isomorphic to Any Commonly Known Fields?

1. What is a field in science?

A field in science refers to a specific area or subject of study. It can also refer to a physical field, such as an electromagnetic field, that exists in a specific region of space.

2. How do scientists determine if something is a field?

Scientists determine if something is a field by conducting research and experiments to gather evidence and data. They also use established criteria and theories to analyze the information and determine if it fits the definition of a field.

3. What are the characteristics of a field?

Some common characteristics of a field include being a specific area or subject of study, having a defined set of principles or laws, and being open to further exploration and discovery.

4. Why is it important to identify something as a field?

Identifying something as a field allows scientists to focus their research and efforts on a specific topic or area of study. It also helps to establish a foundation for further research and understanding in that particular field.

5. Can something be both a field and a subfield?

Yes, something can be both a field and a subfield. A subfield is a specialized area within a broader field. For example, biology is a field, while microbiology is a subfield within biology.

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