Notation for fields and the elements of the sets used to define them.

Question

Context:

While trying to figure out how to prove that the characteristic of a field $\mathcal{F}$ is either $0$ or prime, I realized that I was conflating the elements of the field with integers, because of the notation.

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Question:

Because articulating this question abstractly is beyond my ability, I'll illustrate my confusion with an example:

If $\mathbb{Z}_5$ is the set $\{0,1,2,3,4\}$, along with modular addition and multiplication (i.e. in this case, after performing the standard addition or multiplication, the least remainder after ordinary division by 5; e.g. $4\cdot4 = 1$ and $3+4 = 2$) defines a field. Here, $$ 4 + 1 = 0, $$ i.e. $1$ is the additive inverse of $4$.

Now, all the classical definitions I've seen usually define the additive inverse as follows (this example is taken from Wikipedia):

for every a in F, there exists an element in F, denoted −a, called the additive inverse of a, such that a + (−a) = 0.

Does the phrase "denoted -a" imply that, in the example provided above, we should denote "$1 = -4$"? Or are such definitions simply stating what they mean when they use the symbol (i.e. it is simply saying we should think of $1$ as the additive-inverse-of-4)? In other words, is "$-4$" an element of this field, even if it is not an element of the set used to define the field (does it even make sense to talk of "an element of a field" in this way)?

Just to be clear, if we were to instead denote the additive inverse of a by na, is it then appropriate to write something like $1 = n4$, for the above field? While I can understand what is meant by this, what bothers me is that $n4$ is not an element of any set described or defined, elsewhere.

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Final Considerations

Up until here, I'd be tempted to more or less ignore the whole issue; however, while trying to work out a proof that a field's characteristic is either zero or prime, I found myself wanting to use integers to describe the operations, regardless of whether or not the elements of the set were, themselves, numbers. For example, given the field $\mathbb{F}$ defined by: $$ \left\{ \begin{array}{lll} \mathbb{S} = \{a, b\}, & \begin{array}{c|c|c} \pmb{+} & \pmb{a} & \pmb{b} \\ \hline \pmb{a} & a & b \\ \hline \pmb{b} & b & a \end{array}, & \begin{array}{c|c|c} \pmb{\cdot} & \pmb{a} & \pmb{b} \\ \hline \pmb{a} & a & a \\ \hline \pmb{b} & a & b \end{array}, \end{array} \right. $$

it seems natural and useful to define an expression such as $a + a + a$ as $3a$; however, I'm not sure how to think about the relationship of the symbols introduced by this notation to the field itself (this becomes even more confusing if I use "1" to denote the multiplicative identity). This seems -- at least to my confused mind -- somehow related to the confusion expressed in the first example, above.

Is the use of 3 in "3a" and -4 in "1 = -4" nothing more than a convenient notation to describe the elements of the field? While I feel like that is all they are, I also feel that they carry with them some sort of implied structure (perhaps that of an abelian group?), that isn't accounted for, formally / explicitly.

Answer 1

Several comments.

tl;dr You are overthinking this.

When you write $\mathbb{Z}_5$ as $\{0, 1, 2, 3, 4\}$ those elements are not integers. They are just the names you have chosen for the five elements of that field. Of course those symbols also name integers, and the coincidence is not an accident. The coincidence together with the idea of "modulo $5$" tells you how to do the arithmetic in that field.

The wikipedia definition tells you that a standard way to name the inverse of the element $a$ is to write $-a$. Then in $\mathbb{Z}_5$ you write $-1 = 4$ since $1 + 4 = 0$. So "$-1$" is just another name for the element names "$4$". It's not some other element of the field you forgot to write down.

Every field has a multiplicative identity which is always called $1$ although it is not the integer 1. Then by repeated addition, for every positive integer every field contains an element you could reasonably call $n$. If those elements are all different from one another then the field contains a copy of the integers (even a copy of the rationals) and is said to have characteristic $0$. If there are repeats in the sequence of field elements named by the positive integers then ...

Answer 2

For every a in F, there exists an element in F, denoted −a, called the additive inverse of a, such that a + (−a) = 0.

This actually means: For every $a \in F$ there exists a unique element $b \in F$ such that $a+b=0$. Since the element $b$ is unique for a given $a,$ the set $\{ (a,b) \in F \times F \mid a+b=0 \}$ is a function (I hope that you know the set-theoretic definition of function) which will be denoted with a prefix minus sign, i.e. $-a$ denotes the unique element in $F$ such that $a+(-a)=0.$

For example, in $\mathbb{Z}_5 = \{ 0,1,2,3,4 \}$ (with operations modulo 5) the notation $-3$ denotes the element $2$ since $3+2=0.$ It does not really denote the negative integer $-3$ which is not in $\{0,1,2,3,4\}.$

Is the use of 3 in "3a" and -4 in "1 = -4" nothing more than a convenient notation to describe the elements of the field?

That is correct.

We define a mapping $\mathbb{N}\times F \to F,$ given by $(n,x) \mapsto \underbrace{x+\cdots+x}_{n\text{ copies of }x}.$ For simplicity, and since there's no risk of confusion, as KCd points out, we denote this with $nx$.

For example, in $\mathbb{Z}_5,$ we have $7x = x + x + x + x + x + x + x$. It can be shown that this equals $2x = x+x,$ which in turn equals $2x$ as the product of the elements $2$ and $x$ in $\mathbb{Z}_5.$ (Thus, there is no risk of confusion.)

This notation can be extended to negative $n \in \mathbb{Z}$ by $nx=-(-n)x,$ where the inner minus sign is ordinary negative on the integers, and the outer/front minus sign is negative in $F$.

For example, in $\mathbb{Z}_5,$ we have $(-7)x=-(7x)$ by definition, and therefore can skip the parentheses: $-7x$ is unambiguous.

Answer 3

The notation $-a$ means "the element $b$ that fits the equation $a + b = 0$". So in the integers modulo $5$, $-1 = 4$, $-2 = 3$, $-3 = 2$, $-4 = 1$, and $-0 = 0$.

It is perfectly fine to write $na$, for a positive integer $n$, to mean the sum of $a$ with itself $n$ times. There is no inconsistency between the "$3$" in $3a$ as $a + a + a$ and the "$3$" in $3a$ as the product of the element $3$ in the field and $a$ because $$ a + a + a = 1\cdot a + 1 \cdot a + 1 \cdot a = (1 + 1 + 1)\cdot a = 3a. $$

By the way, it is a terrible idea to limit your viewpoint of the integers modulo 5 as being only the symbols $0,1,2,3,4$ with a wraparound addition and multiplication rule modulo $5$. You should accept $7$, $11$, and $-46$ as perfect reasonable integers modulo $5$ that happen to be the same thing in the integers modulo $5$ as $2$, $1$, and $4$, respectively.

After all, you should be perfectly comfortable with writing the same fraction in many ways: $1/2 = 2/4 = 3/6 = (-6)/(-12)$, and so on. It would be extremely inconvenient if you were only allowed to work with fractions in reduced form (try to define addition and multiplication when the inputs and the output can only be written as reduced form fractions). Similarly, it is terrible to restrict yourself to only working with the integers modulo $5$ as $0,1,2,3,4$. The integers modulo $5$ can be represented by all integers, but there is a specific rule about when they get identified: $a$ and $b$ are the "same integer mod $5$" when $a - b$ is a multiple of $5$, just like the fractions $a/b$ and $c/d$ are the "same fraction" when $ad = bc$ as integers.