# What is the difference between multiplicative group of integers modulo n and a Galois Field

What is the difference between multiplicative group of integers modulo n and a Galois Field?

Is $\mathbb{Z}^*_p$ with p prime the same as $GF(p)$? Or is it the same as $\mathbb{Z}/n\mathbb{Z}$?

Sorry for the short and simple question, but this bit of notation is confusing me, so a clear different should help me alot!

Thanks!

• A galois / finite field is a field, not a multiplicative group. Do you know the definition of "group," "ring" and "field"? But (when $p$ is prime) $GF(p)$ and ${\bf Z}/p{\bf Z}$ are the same, also sometimes denoted ${\bf F}_p$. For composites $n$, the integers mod $n$, denoted ${\bf Z}/n{\bf Z}$, makes sense but unless $n$ is a prime power, there is no field with $n$ elements. If $q=p^f$ is a prime power, then there is exactly one field with $q$ elements, denoted $GF(q)$ or ${\bf F}_q$. The notation should not be too confusing if you consult the definitions of the terms!
– anon
May 31, 2013 at 14:13
• Yes, i know the definition of those terms. Thank you for the clear response! May 31, 2013 at 14:26

If $p$ is prime then the ring $\mathbb Z/p\mathbb Z$ is in fact a field and is "the" field with $p$ elements (as there is only one up to isomorphism). Note however that for prime powers $q=p^n$ with $n>1$, the ring $\mathbb Z/q\mathbb Z$ is not a field, hence is different from the field $GF(q)$ (or $\mathbb F_q$, depending on author).
• So $\mathbb{Z}^*_p$ is the same as $\mathbb{Z}/n\mathbb{Z}*$? Thanks for your help! May 31, 2013 at 14:11
• @SanderDemeester Is $p$ the same as $n$?
• @Sander: $\mathbb{Z}_p^*$ denotes the multiplicative group of non-zero elements of $\mathbb{Z}_p$. May 31, 2013 at 17:35