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?
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1Do you mean "the" finite field of order 4? Because $\mathbb{Z}/4\mathbb{Z}$ is not a field, so they're definitely different. – Randall May 02 '18 at 15:56
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Yes sorry, I guess F is a field – John Cataldo May 02 '18 at 15:57
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They line up if and only if $n$ is a prime. – Randall May 02 '18 at 15:57
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And how can it be a field if 2 has no multiplicative inverse? – John Cataldo May 02 '18 at 15:57
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It's not. There IS a field of order 4, but it is not $\mathbb{Z}_4$. – Randall May 02 '18 at 15:58
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Where do they not line up in this case where n=4? – John Cataldo May 02 '18 at 15:58
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So which field is it? – John Cataldo May 02 '18 at 15:59
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@StanislasHildebrandt There is a unique $\mathbb{F}_{4}$ up to isomorphism, and it is the Galois Field GF(4). – Frenzy Li May 02 '18 at 16:02
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You get it typically from forming a quotient of $\mathbb{Z}_2[x]$ modulo an irreducible quadratic. – Randall May 02 '18 at 16:03
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@StanislasHildebrandt $ \mathbb{Z}_2[x]/(x^2+x+1) $ is the $ \mathbb{F}_4 $. – Bach May 02 '18 at 16:04
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Groups and a field are not unique up to the number of elements. You can have multiple different groups and a field with $n$ elements. $\mathbb Z_n$ is a finite cyclic group and is a field only if $n$ is prime. $Z_4$ is not a field because $22 = 0$ which is a zero divisor and there is not inverse $a$ so that $2a = 1$. $\mathbb F_4={0,1,a,b}$ where $a+1=b;b+1=a$ and $aa = b;ab=1;bb=a$. – fleablood May 02 '18 at 16:16
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$ \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} $.
Bach
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$\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 $d$.
Randall
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1Well, it's a lot to bite off. If the irreducible $g(x)$ has degree $d$ then every coset $f(x) +I$ of the quotient ring can be uniquely written--using long division---as $b_0 + b_1x + \cdots + b_{d-1}x^{d-1} + I$. Now count up the number of ways to choose the coefficients and you get $p^d$ different cosets/elements. – Randall May 02 '18 at 16:35
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