📂Abstract Algebra

Solid State Physics

Buildup

For any element $r$ of a ring $R$ that satisfies $n \cdot r = 0$, the largest natural number $n$ is defined as the Characteristic of $R$. If such a natural number does not exist, $0$ is defined as the characteristic of $R$. A ring with a multiplicative identity, that is, a unit element, has the following properties:

• [1]: If the characteristic of $R$ with a unit element is $n>1$, then $R$ has a subring isomorphic to $\mathbb{Z}_{n}$.
• [2]: If the characteristic of $R$ with a unit element is $0$, then $R$ has a subring isomorphic to $\mathbb{Z}$.

Similarly, a field $F$ has the following properties for a prime $p$:

• [1]’: If the characteristic of $F$ is $p$, then $F$ has a subfield isomorphic to $\mathbb{Z}_{p}$.
• [2]’: If the characteristic of $F$ is $0$, then $F$ has a subfield isomorphic to $\mathbb{Q}$.

Definition ¹

Here, the integer field $\mathbb{Z}_{p}$ and the rational field $\mathbb{Q}$ are called Prime Field.

Description

As the term Prime suggests, it’s an extremely important field.

Considering the converse of [1]’ and [2]’, if there is no subfield that makes $F$ isomorphic to these prime fields, then $F$ is not a field. Therefore, it can be useful in determining whether something is a field, especially since it is familiar to us.

The characteristic might be a little confusing with nilradical, but they are concepts related to addition $\displaystyle \sum_{i=1}^{n} r = nr = 0$ and multiplication $\displaystyle \prod_{i=1}^{n} a = a^{n} = 0$ respectively. Also, the characteristic is concerned with the smallest $n$ that satisfies a condition, while nilradical is interested in a $a$ that satisfies a condition.