I was wondering as I read about characteristic of a ring: Is there an infinite ring with nonzero characteristic? We have $1+1+\ldots+1=0$, but that doesn't seem to imply that the number of elements in the ring is finite.
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cf. http://math.stackexchange.com/questions/356649/can-a-ring-of-positive-characteristic-have-infinite-number-of-elements – user66081 Jan 15 '14 at 12:29
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1It is $\Bbb Z/p\Bbb Z$ as in the field of p elements $F_p$. The polynomial ring $F_p[x]$ has infinitely many elements, but characteristic p. – rschwieb Jan 15 '14 at 12:41
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1These are the polynomials with coefficients in $\mathbf{Z}/p\mathbf{Z}$. Other examples are the polynomial rings K[X] in X over any finite field K. – gammatester Jan 15 '14 at 12:42
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Thanks @rschwieb and gammatester – Kunal Jan 15 '14 at 12:43
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The algebraic closure of any finite field is infinite. – Tim Seguine Jan 15 '14 at 12:55
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Sure, like the polynomial rings $F[x]$ where $F$ is a finite field. Then we also have $F(x)$ and $F[[x]]$. The list goes on . . ..
Hope this helps. Cheers,
and as always,
Fiat Lux!!!
Robert Lewis
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