Field Theory in Abstract Algebra 📂Abstract Algebra

Field Theory in Abstract Algebra

Definition ¹

If a ring $(R , + , \cdot)$ has an identity element $1 \in R$ for multiplication $\cdot$ , $1$ is called a unity.
In a ring $R$ with a unity, the element $r \ne 0$ that has a multiplicative inverse is called a unit.
If every element other than $0$ is a unit in a ring $R$ with a unity, it is called a division ring.
A division ring $R$ that is commutative with respect to multiplication is called a field.

Explanation

In short, a field $(F , + , \cdot )$ is a commutative ring $0 \in F$ in which every non-zero element has a multiplicative inverse. While it may seem complex when thinking abstractly in terms of algebra, considering the real number space $\mathbb{R}$ that one learns about in analysis actually illustrates how natural this ‘algebraic structure’ actually is.

Why Elements with Inverses are Called Units

Although the term “unity” for a unity element is easy to accept in English, many people find it perplexing as to why elements with inverses are called “units.” Usually, the word “unit” is translated to mean “unit” in the sense of a standard measure, which often seems unrelated to the concept of inverses. So why define these elements as “units”?

Here’s an interesting conjecture to consider. When the field of algebra was developing, much of the research was focused on integers. The fact that we write the set of integers as $\mathbb{Z}$ stems from the German word “Zahlring,” where “Zahl-” means “number,” and “-ring” is translated as ring. It is not hard to accept that many concepts used in algebra originated from number theory.

Let’s consider the integer ring $\mathbb{Z}$ .

$\mathbb{Z}$ contains infinitely many integers like $\cdots , -2 , -1, 0, 1 , 2 ,\cdots$ . The only element that acts as an identity under multiplication here is $1$ , and the elements that have inverses are only $-1$ and $1$ . If you’re familiar enough with mathematics that you can study abstract algebra, the fact that $-1$ and $1$ are called “units” won’t seem strange. Given this background, it might have been fitting to extend the term “unit” to similar elements when considering various algebraic structures beyond integers.

Up to $\mathbb{R}$ , every element excluding $0$ has a multiplicative inverse $\displaystyle {{1} \over {r}} \in \mathbb{R}$ , meaning every element except $0$ is a unit. When you think about it, there’s no reason not to consider $r \ne 1$ as a unit since you can multiply some number $a$ by it to get the number you want $x$ . And this number $a$ is obvious when $a = r^{-1}x$ thrives, but it is not guaranteed without $r^{-1}$ .