What is the difference between $\Bbb F_4$ and $\Bbb Z/4\Bbb Z$

Question

And more generally what is he difference between $\Bbb F_n$ and $\Bbb Z/n\Bbb Z$ when we look at them as rings with multiplication and addition? Aren’t the tables of multiplication and addition the same?

Answer 1

$ \mathbb{F}_4 $ means the finite field with $ 4 $ elements, but $ \mathbb{Z}/4\mathbb{Z} $ is not a field, since $ \bar 2\times \bar2=\bar 0 $ in $ \mathbb{Z}/4\mathbb{Z} $.

Generally, $ p $ is prime if and only if $ \mathbb{Z}/p\mathbb{Z} $ is a field and the only field with $ p $ elements up to isomorphism. Otherwise, any finite field with $ q=p^r $ elements where $ r>1 $ is not isomorphic to the quotient ring of $ \mathbb{Z} $.

Answer 2

$\mathbb{F}_n$ is typical notation for "the" finite field of $n$ elements. It is a theorem that such a thing is unique up to isomorphism. The notation $GF(n)$ is also used. Also, $n$ must be a power of a prime $n=p^d$ so you may also see $GF(p^d)$. (Consequence of another theorem: $GF(p^d)$ is vector space of dimension $d$ over $\mathbb{Z}_p$.)

To your question, $\mathbb{Z}_p$ is a field when $p$ is prime, so by the uniqueness result, $GF(p)$ "is" $\mathbb{Z}_p$. When $n$ is NOT prime, $\mathbb{Z}_n$ is certainly not a field, so it cannot be $GF(n)$. To see this, if $n$ is not prime then $n = ab$ for integers $1< a< b <n$. But then $[a][b]= [n] =[0]$ in $\mathbb{Z}_n$ so we have zero divisors. Thus $\mathbb{Z}_n$ is not a field.

To get $GF(p^d)$ when $d > 1$ you need to know a good deal about ideal theory and polynomials. There is a theorem that $\mathbb{Z}_p[x]/I$ is a field when $I$ is a maximal ideal in $\mathbb{Z}_p[x]$. You then classify all such maximal ideals $I$ as the principals $I = (g(x))$ where $g(x)$ is irreducible. A simple count then shows that $\mathbb{Z}_p[x]/I$ has order $p^d$ where $I = (g(x))$ and $g(x)$ has degree $d$.