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I read some books about finite fields, sometimes the author refers to the finite field $\mathbb{F}_p$ and sometimes to the finite cyclic group $\mathbb Z_p$. What is the difference between them?

Squirtle
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Gold Rose
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  • Well, for one $\mathbb{Z}_0$ is called the trivial addition group, while $\mathbb{F}_0$ does not exist (because $0\neq 1$ for fields http://en.wikipedia.org/wiki/Field_with_one_element) – Squirtle May 29 '14 at 01:59

1 Answers1

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Roughly speaking, they are the same set but with different emphasis.

If we talk about the finite field $\Bbb F_p$, where $p$ is a prime, this can be visualised as the integers $\{0,1,\ldots,p-1\}$ with the two operations of addition modulo $p$ and multiplication modulo $p$. You can check that the field axioms hold true in this case.

If we talk about the finite cyclic group $\Bbb Z_p$ then we are still visualising the numbers $\{0,1,\ldots,p-1\}$, but we are only working with one operation, which would be addition modulo $p$. You can check that the group axioms are satisfied, and moreover that the group is cyclic.

Notice however that this only works if $p$ is prime. A finite field with $n$ elements $\Bbb F_n$ exists if $n$ is a power of a prime, $n=p^\alpha$, but if $\alpha>1$ then this is not the same as $\Bbb Z_n$.

The finite cyclic group $\Bbb Z_n$ exists for all positive integers $n$.

Examples:

  • The finite field $\Bbb F_{31}$ and the cyclic group $\Bbb Z_{31}$ consist of the same numbers: the only real difference is that in $\Bbb F_{31}$ we consider problems involving multiplication and in $\Bbb Z_{31}$ we don't.
  • There is a finite field $\Bbb F_{32}$ and a cyclic group $\Bbb Z_{32}$, but they are not at all the same.
  • There is a cyclic group $\Bbb Z_{33}$, but there is no field $\Bbb F_{33}$ because $33$ is not a power of a prime.
David
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  • OK thank you mr David. you say Zp is working on one operation addition modulo p. If finite cyclic group refer by Zp*. in this case working on multiplication modulo p. Is this correct or not? – Gold Rose May 29 '14 at 01:50
  • That's correct, except that $0$ must be omitted: $\Bbb Z_p^={1,2,\ldots,p-1}$. If $p$ is not prime other elements must be omitted too, e.g., $\Bbb Z_8^={1,3,5,7}$. – David May 29 '14 at 01:51
  • The "'same' numbers?" I'm not sure what that means. $\mathbb Z_n$ is not just a cyclic group, it is often used for the full ring, and that ring is a field when $n$ is prime... – Thomas Andrews May 29 '14 at 02:30
  • @ThomasAndrews sure, but the OP is not asking about rings. "The same numbers". . . well it just means the same numbers! The elements of $\Bbb F_{31}$ can be thought of as ${0,1,\ldots,30}$, and the elements of $\Bbb Z_{31}$ as ${0,1,\ldots,30}$. – David May 29 '14 at 03:09
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    The OP is not talking about additive groups, either. My point is, your first bullet at the end implies $Z_p$ is just an additive group. It is not. $\mathbb Z_n$ is a ring. The elements of $\mathbb Z_{31}$ are not ${0,1,2,\dots,30}$ but co-sets of which these numbers are representatives. It's basically just formally wrong and potentially confusing to say they contain the same elements. – Thomas Andrews May 29 '14 at 03:13
  • @ThomasAndrews starting with your last point: I completely disagree, on educational grounds. I know perfectly well that we are talking about cosets. But if you read the OP as asked, I believe it is quite clear that giving an answer at that level would be very confusing and would not give any useful assistance. You will have noticed, however, that in my answer I did not say that the set is integers but that it can be visualised as integers: I put it that way in order to address exactly this point without confusing the OP. – David May 29 '14 at 04:00
  • @ThomasAndrews With regard to your first point, the OP referred to "the finite cyclic group $\Bbb Z_p$". As there is up to isomorphism only one cyclic group of any finite order, the operation can be visualised as addition modulo $p$; and once again, at the level indicated by the question it is not unreasonable to say that it is addition modulo $p$. And for the middle, I said $\Bbb Z_p$ is an additive group, which it is. I did not anywhere say it is just an additive group. To repeat: I did not address the issue of rings because it wasn't asked. – David May 29 '14 at 04:00
  • when multiplication operation use for finite field Fp* and finite cyclic group Zp*.Can we say that same when P is prime number. Is this comment correct or not? – Gold Rose May 29 '14 at 18:25
  • If $p$ is prime, the finite field $\Bbb F_p$ (not $\Bbb F_p^$) and the cyclic group $\Bbb Z_p^$ both have multiplication modulo $p$. – David May 30 '14 at 00:09