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Math Help - Finding Prime Fields of the Ring Z_3 x Z_5

  1. #1
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    Finding Prime Fields of the Ring Z_3 x Z_5

    I have a ring R = Z_3 x Z_5 and I have to find its two prime fields.

    I have tried to find out what these minimal rings could be. However, if the ring must have elements (0,0) and (1,1), then it would also have to have elements a - b according to the subring criterion, which would cause the ring to have (0,0) - (1,1) = (-1,-1) = (2,4). Now also (2,4) - (1,1) would have to be in subring S, so eventually this would make the whole ring R. This leads to the fact that R has no subrings, which makes me confused, since the task is to find two of them.

    If I make the assumption that the ring does NOT need to have (1,1) however, then I could get two subrings (which are also prime fields, I think?):
    S1 ={ (0,0), (0,1), (0,2), (0,3), (0,4) } and
    S2 = { (0,0), (1,0), (2,0) }

    Can I assume that the subrings do not have (1,1)?
    Are S1 and S2 prime fields of Z_3 x Z_5 ?

    If not, then some advice would be more than welcome.
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  2. #2
    Senior Member roninpro's Avatar
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    Could you please state your definition of "prime field"? I looked at a few sources and they all use "prime field" with respect to some field. Your ring \mathbb{Z}_3\times \mathbb{Z}_5 is not a field, so it is not clear how to approach your problem.
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  3. #3
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    Definition: a field is a prime field if it has no proper subfields.

    I am aware that \mathbb{Z}_3\times \mathbb{Z}_5 is not a field, but I suppose there some of its subrings can be fields.
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  4. #4
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    Quote Originally Posted by Koaske View Post
    Definition: a field is a prime field if it has no proper subfields.

    I am aware that \mathbb{Z}_3\times \mathbb{Z}_5 is not a field, but I suppose there some of its subrings can be fields.

    Oh, so that's what you meant....well, then it's easy: both \mathbb{Z}_3\times \{0\}\,,\,\,\{0\}\times\mathbb{Z}_5 are

    subfields though not subrings if you require that a subring has the same unit as the whole ring.

    Tonio
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  5. #5
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    So those are the same
    S1 = { (0,0), (0,1), (0,2), (0,3), (0,4) } and
    S2 = { (0,0), (1,0), (2,0) }
    that I managed to find (though they aren't subrings, so my naming might be a bit misleading).

    Thanks for confirming this!
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