are the integers modulo 4 a field?

Question

Basically are the integers mod 4 a field? I want to know because I am reading a text and it has a problem assuming the integers modulo any number are a field

Answer 1

It is not a field, because $2$ as no inverse for the multiplication law:

$$2\times 0=0,\quad 2\times 1=2,\quad 2\times 2=0,\quad 2\times 3=2.$$

The integers mod $n$ will be a field if, and only if, $n$ is a prime number.

Answer 2

No. Addition and multiplication mod $n$ are well defined, so $\mathbb{Z}/n$, the integers mod $n$, is always a ring, but not a field in general unless $n$ is a prime. In particular, the integers mod 4, (denoted $\mathbb{Z}/4$) is not a field, since $2\times 2 = 4 = 0\mod 4$, so $2$ cannot have a multiplicative inverse (if it did, we would have $2^{-1}\times 2\times 2 = 2 = 2^{-1}\times 0 = 0$, an absurdity. $2$ is not equal to $0$ mod $4$). For this reason, $\mathbb{Z}/p$ a field only when $p$ is a prime.

(Note that sometimes people denote the ring of integers mod $n$ as $\mathbb{Z}_n$, but I prefer not to, and reserve that notation for the $p$-adic integers. So I prefer to only write it as $\mathbb{Z}/n$ (or $\mathbb{Z}/(n)$ or $\mathbb{Z}/n\mathbb{Z}$ if you're not into the whole brevity thing). However if you're more familiar with the $\mathbb{Z}_n$ notation, feel free to substitute.)

However, while $\mathbb{Z}/4$ is not a field, there is a field of order four. In fact there is a finite field with order any prime power, called Galois fields and denoted $\mathbb{F}_q$ or $GF(q)$, or $GF_q$ where $q=p^n$ for $p$ a prime. If $q$ is not a prime power, say a composite with distinct primes like $6=2\times 3,$ then there is no field of order $q$.

But $\mathbb{F}_q$ is not $\mathbb{Z}/q$ unless $n=1$ so $q=p$ is itself a prime. So $\mathbb{Z}/2$ and $\mathbb{F}_2$ are the same ring, but $\mathbb{Z}/4$ and $\mathbb{F}_4$ are not. Nor are $\mathbb{Z}/8$ and $\mathbb{F}_8.$ Similarly, $\mathbb{Z}/3$ and $\mathbb{F}_3$ are the same ring, but $\mathbb{Z}/9$ and $\mathbb{F}_9$ are not.

So then what is $\mathbb{F}_4$, the finite field of order four? Well wikipedia has some details, but one construction in brief is $\mathbb{F}_4 = \mathbb{F}_2[x]/(x^2+x+1) = \{0,1,x,x+1\}$. So it's residues of polynomials with coefficients in $\mathbb{F}_2$ modulo an irreducible quadratic (of which $x^2+x+1$ is the only one).

One needs to ask whether these rings are isomorphic, meaning the same rings dressed up in different notations. After all they both have four elements, and being the same size is a requirement for isomorphism. But they are not isomorphic.

Note that the characteristic (number of times you add one to itself to get zero) of $\mathbb{Z}/4$ is 4, while the characteristic of $\mathbb{F}_4$ is 2. So all elements of $\mathbb{F}_4$ are of order 2 under addition, while $\mathbb{Z}/4$ has two elements of order 4. The rings' additive groups are not isomorphic.

The multiplicative group of nonzero elements of $\mathbb{F}_4$ is the cyclic group of three elements, whereas the nonzero elements of $\mathbb{Z}/4$ do not form a group at all, since they are not closed under multiplication. In $\mathbb{Z}/4$, $2\times 2 = 0$, whereas in $\mathbb{F}_4$, $x\cdot x=x^2=x+1$ and $(x+1)\cdot (x+1)=(x+1)^2=x^2+2x+1 = (x^2+x+1)+x = x.$ So there is no nilpotent element in $\mathbb{F}_4$, as we discussed in the first paragraph cannot happen in a field. To say it another way, the group of units of $\mathbb{Z}/4$ has cardinality two, $(\mathbb{Z}/4)^\times=\{1,3\},$ while the group of units of $\mathbb{F}_4$ has cardinality three, $\mathbb{F}_4^\times=\{1,x,x+1\},$ and so they are not isomorphic.

So neither the underlying additive groups, nor the multiplicative monoids of these rings are isomorphic, so of course the rings are not isomorphic.

Answer 3

If $q=p^n$, then $\mathbf F_q$ denotes the field with $q$ elements.

You have to know that for any integer $n\ge 1$, there exists a finite field with $p^n$ elements, and this field is unique up to an isomorphism.

It is even unique in the still more restrictive sense: a field with $p^n$ elements is unique within a given algebraic closure of the prime field $\mathbf F_p$. Furthermore, for any two such finite fields, $$\mathbf F_{p^m}\subseteq\mathbf F_{p^n}\iff m\mid n.$$