Can a field contain 28 elements? (please check my answer)

[jacobsldr]

The question: Can a field contain 28 elements?

I want to answer: No, since I can create an isomorphism to Zmod28, and since 28 is not prime I know it is not a field. Therefore a field can't contain 28 elements.

Does this seem to be a sufficient response to the question?

Thanks for any help you can provide.

 

[stapel]

Is it safe to assume that "28 elements" means "exactly 28 elements"? (That is, no more and no less than 28 elements.)

 

[Florian]

Yes, a field can contain 28 elements! But there is just one field with 28 elements. It may be obvious that 28 is not a prime, but 28 = 3^3.
For every prime number p and positive integer n, there exists a finite field with p^n elements.

But have fun making the addition- and multiplication-table.

greetings

 

[jacobsldr]

Is it safe to assume that "28 elements" means "exactly 28 elements"? (That is, no more and no less than 28 elements.)

Yes, I believe the intent of the question (however, I reread it and your right it is vague on that) is exactly 28 elements.

Yes, a field can contain 28 elements! But there is just one field with 28 elements. It may be obvious that 28 is not a prime, but 28 = 3^3.
For every prime number p and positive integer n, there exists a finite field with p^n elements.

But have fun making the addition- and multiplication-table.

greetings

Well 3^3=27, but I think I see your point. So if I can create a p^n the equals the number of elements I can say it is a field?

Thanks for the replys!!!

PS: no thanks on the add/mult table, I did one last quarter by hand for the symmetrical group S4 for a project (it was 500+ elements) don't even want to think about a 28 element table.

 

[daon]

The question: Can a field contain 28 elements?

I want to answer: No, since I can create an isomorphism to Zmod28, and since 28 is not prime I know it is not a field. Therefore a field can't contain 28 elements.

Does this seem to be a sufficient response to the question?

Thanks for any help you can provide.

Stating you can construct an isomporphism does nothing. You must define a function and prove it is an isomorphism unless you are given some theorem(s) which state you can do so without creating one. This is all on the basis that you "know" Z_28 is a Ring but not a Field.

 

[DrMike]

The question: Can a field contain 28 elements?

I want to answer: No, since I can create an isomorphism to Zmod28, and since 28 is not prime I know it is not a field. Therefore a field can't contain 28 elements.

Does this seem to be a sufficient response to the question?

Thanks for any help you can provide.

You'd have to prove that you can create the isomorphism. And the first problem you come up with is - from what? What properties of your F[sub:30nzg59q]28[/sub:30nzg59q] ensure that it is isomorphic to Z[sub:30nzg59q]28[/sub:30nzg59q]? For the proof to work, this list of properties must be a subset of the properties of a field.