Prove that Characteristic of a Field is Prime or 0

The characteristic of a field is the smallest positive integer n such that nI = 0 where I denotes the multiplicative identity. If no such n exists, then the characteristic is zero. It is denoted by the symbol char(F). In this article, the characteristic of a field is either 0 or a prime number.

Characteristic of a Field is 0 or Prime

Proof:

Let F be a field.

So F is a commutative ring with unity I.

If there is no least positive integer n such that nI=0, then char(F)=0.

So suppose that char(F) = n. This means n is the least positive integer such that nI=0. We need to show that n is a prime number.

If possible suppose that n is a composite number. So we can write

n = pq

for some integers p and q with 1< p, q < n. Now nI = 0 implies that

(pq)I = 0

⇒ (pI) (qI) = 0

As every field is an integral domain and an integral domain contains no zero divisor, this implies that

either pI = 0 or qI = 0.

This contradicts the fact that n is the smallest positive integer such that nI = 0. So our assumption is wrong. This shows that n=char(F) must be a prime number. This completes the proof of the fact that the characteristic of a field is either a prime or 0.

Main Topic: Field Theory: definition, Examples, Theorems

Fields of characteristic 0

The field of real numbers and complex numbers, that is, ℝ and ℂ are fields of characteristic zero. Also, the field ℚ of rational numbers has characteristic 0. Therefore, every algebraic number field (a field extension of ℚ) is of characteristic 0.

The p-adic fields are of characteristic 0. These fields are extensively used in modern number theory now a days. For example, the 2-adic field ℚ₂ has characteristic zero.

Also Read: Integral Domain | Characteristic of a Ring

Prove that every finite integral domain is a field

Fields of Prime Characteristic

If a field is of prime characteristic, say p, then the field must be a finite field. Some examples of fields of prime characteristic are listed below.

• The field 𝔽_pⁿ of elements pⁿ. Note that 𝔽_pⁿ ≅ ℤ/pⁿℤ.
• The field GL_n(𝔽_pⁿ) of non-singular matrices over 𝔽_pⁿ.

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