I have already taken a look at this answer. Somehow it did not answer my question.
As I can find, in various literatures,
- A lecture note, Definition 4.1:
Let
$F$
be a field. A subset
$K$
that is itself a field under the operations of
$F$
is called a
subfield
of
$F$. - Another lecture note, Section 7.4.2: A subfield $G$ of a field $F$ is a subset of the field that is itself a field under the operations of $F$.
Now, if we consider the operations of the field to be $+ \bmod n$ and $\times \bmod n$. We find that $\mathbb{Z}_2$ and $\mathbb{Z}_5$ are both fields under these operations.
But in order to get a good feeling of subfields, we try to consider $\mathbb{Z}_{3^2} = \mathbb{Z}_9$. We find that this not a field under the afore stated operations.
Not all the non-zero elements, notably 3 and 9 ($\gcd(3,9) \not=1$ and $\gcd(6,9) \not=1$), do not have multiplicative inverses.
Indeed, as Wikipedia states,
Even though all fields of size $p$ are isomorphic to $\mathbb{Z}/p\mathbb{Z}$, for $n \ge 2$ the
ring $\mathbb{Z}/p^n\mathbb{Z}$ (the ring of integers modulo $p^n$) is not a field. The
element $p$ $(\bmod\ p^n)$ is nonzero and has no multiplicative inverse.
Looking for examples, we find one here for $GF(2^3)$. This is based on polynomials.
Now, coming to my original point on (understanding) subfield or prime subfield
of finite fields, please tell me,
- Whether it is totally impossible to construct purely numerical examples of
fields of size $p^n$. - Given a (non-numerical) field of size $p^n$, (one can be found in page 90 (16) of this document), what is the best way to identify the subfield(s) and prime subfield? I appreciate an answer which nurtures my intuition, not a theoretical one which puts me deep in difficult mathematical terms.
Best Answer
Let me address your first question. First, I want to argue that there is no precise meaning of "involving numbers only". For example, given a finite field $F$ of size $4$ constructed in the usual manner (quotient of a polynomial ring over $\mathbb{Z}/2\mathbb{Z}$), I can choose a set of numbers, say $$S=\{37,\tfrac{5}{19},\pi,e\}$$ and, choosing a bijection of $S$ with $F$, use transport of structure to give $S$ the structure of a field. The field structure does not depend in any way on what the underlying set is "made of".
However, along the lines of what I think you are ultimately after, you can obtain finite fields of any possible order using larger rings of integers. For example, $\mathbb{Z}[i]/(3)$ is a finite field of size $9$, and $\mathbb{Z}[i]$ consists of very reasonable numbers, $$\mathbb{Z}[i]=\{a+bi\mid a,b\in\mathbb{Z}\}.$$
Now let me addressr your second question. Let's use $\mathbb{F}_p$ to mean $\mathbb{Z}/p\mathbb{Z}$, a finite field of order $p$ - it is a very common notation that is slightly less cumbersome, but doesn't mean anything different, they are exact synonyms.
A finite field of order $p^n$ is often constructed by taking the polynomial ring $\mathbb{F}_p[x]$, choosing an irreducible polynomial $f\in \mathbb{F}_p[x]$ of degree $n$, and then making the field $$F=\mathbb{F}_p[x]/(f).$$ Now, the division algorithm for polynomials tells you that each equivalence class in this quotient can be uniquely identified by a representative of degree $<n$. In other words, $$\begin{align*} F&=\{a_0+a_1x+\cdots +a_{n-1}x^{n-1}+(f)\mid a_0,a_1,\ldots,a_{n-1}\in\mathbb{F}_p\}\\\\ &=\left\{\,\overline{a_0+a_1x+\cdots +a_{n-1}x^{n-1}}\,\;\middle\vert\;a_0,a_1,\ldots,a_{n-1}\in\mathbb{F}_p\right\}\\\\\\ &=\{a_0+a_1\overline{x}+\cdots +a_{n-1}\overline{x}^{n-1}\mid a_0,a_1,\ldots,a_{n-1}\in\mathbb{F}_p\} \end{align*}$$ Letting the symbol $\alpha$ be a stand-in for $\overline{x}$, you can think of $F$ as being $\mathbb{F}_p$ with a new element "$\alpha$" added in, where $\alpha$ is a root of $f$, and you can write $F=\mathbb{F}_p[\alpha]$.
Now, the prime subfield of $F$ is just the "constant" polynomials, i.e. the ones with no $\alpha$'s in them: $$\text{the prime subfield of }F=\{a_0+0\alpha+\cdots+0\alpha^{n-1}\mid a_0\in\mathbb{F}_p\}$$ and for each divisor $d\mid n$, the unique subfield of $F$ of order $p^d$ is the collection of polynomials in $\alpha$ whose terms are those of exponents that are multiples of $n/d$: $$\text{the subfield of }F\text{ of order }p^d=\{a_0+a_1\alpha^{n/d}+\cdots+a_{d-1}\alpha^{(d-1)n/d}\mid a_0,a_1,\ldots,a_{d-1}\in\mathbb{F}_p\}$$ (clearly, the above set has cardinality $p^d$, because it takes $d$ elements of $\mathbb{F}_p$ to specify a given element of the above set, namely, each of the coefficients of the powers of $\alpha$. To see that it is a field, remember that $(a+b)^p=a^p+b^p$ in a field of characteristic $p$.)