Construct a field with all positive intergers.

In summary, the author attempted to create a field that satisfied all basic properties with only and all positive integers, but was unable to do so because he needed fractional and negative numbers. After speaking with a professor, the author realized that he needed to use a different approach and work with a higher order table of values.
  • #1
FallMonkey
11
0

Homework Statement



Is it possible to construct a field, that satisfies all basic propeties (http://en.wikipedia.org/wiki/Field_(mathematics)#Definition_and_illustration , the six bolded part ), with only and all positive integers?

Remeber, you don't have 0/negative numbers/fraction in positive integers, so you may have to redefine addition and multiplication to help.

And, quoted from my professor, "If you rephrase this question in terms of set theory, it would be rather easy then".

Homework Equations



None.

The Attempt at a Solution



OK first I tried finite field approach, and thought that I could be able to make a value table to enumerate all possible values of addition and multiplication, based on that we refer to Galois Extension to construct higher order of the table, until the p^n order of a finite field. But after speaking with professor, it was said to be wrong direction because I'm supposed to use all postive intergers, other than such special finite field case. Obviously what I'm trying to achieve is a infinite field, and I only know three of such, rational/real/complex.

And there I got a remark from him, that I wrote above, "If you rephrase this question in terms of set theory, it would be rather easy then". I'm only begnning in set theory as a junior math major, and I know I could probably be not able to answer it right now. But curiosity is killing me inside and I'd love to know even a little about what I should know for proving or disproving this postulate.

Thank you very much, and let me know if I do not make it clear somewhere.
 
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  • #2
Rephrase the question to: You have a countable set. Can you turn it into a field?
 
  • #3
Office_Shredder said:
Rephrase the question to: You have a countable set. Can you turn it into a field?

That's such amazing rephrase, I suddenly feel I have more for it.

Thanks for the tips, I'll see what I can do now.
 

FAQ: Construct a field with all positive intergers.

What is a field with all positive integers?

A field with all positive integers is a mathematical structure that contains a set of numbers, known as integers, which are positive whole numbers (1, 2, 3, ...). It also includes two operations, addition and multiplication, that follow certain rules such as closure, associativity, and distributivity.

How do you construct a field with all positive integers?

To construct a field with all positive integers, we start with the set of positive integers and define two operations, addition and multiplication, that follow the rules of a field. Addition is defined as the sum of two positive integers, and multiplication is defined as the product of two positive integers. These operations must satisfy the rules of closure, associativity, and distributivity to form a field.

What are the properties of a field with all positive integers?

A field with all positive integers has several important properties, including closure, associativity, commutativity, identity elements, inverse elements, and distributivity. Closure means that the result of an operation between two positive integers is also a positive integer. Associativity means that the grouping of operations does not affect the result. Commutativity means that the order of operations does not affect the result. Identity elements are the numbers 0 for addition and 1 for multiplication. Inverse elements are numbers that, when combined with another number, result in the identity element. Distributivity means that multiplication distributes over addition.

Why is it important to construct a field with all positive integers?

A field with all positive integers is an important mathematical structure because it allows us to perform operations on positive integers in a systematic and consistent way. It also serves as the basis for more complex mathematical structures, such as the real numbers and complex numbers, which are essential in many areas of science and mathematics.

Are there any limitations to constructing a field with all positive integers?

Yes, there are limitations to constructing a field with all positive integers. One limitation is that it cannot include negative numbers or fractions, which are important in many real-world scenarios. Additionally, the set of positive integers is infinite, so it is impossible to list or represent all of the numbers in the field. Another limitation is that some operations, such as division, are not well-defined for all pairs of positive integers in a field, so we must define them separately.

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