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Subfield — Definition, Formula & Examples

A subfield is a subset of a field that is itself a field using the same addition and multiplication. For example, the rational numbers Q\mathbb{Q} form a subfield of the real numbers R\mathbb{R}.

Let (F,+,⋅)(F, +, \cdot) be a field. A subset K⊆FK \subseteq F is a subfield of FF if KK is closed under addition, subtraction, multiplication, and division by nonzero elements, contains the additive identity 00 and multiplicative identity 11, and satisfies all field axioms under the operations inherited from FF.

How It Works

To verify that a subset KK of a field FF is a subfield, you check three conditions: (1) KK is nonempty (it must contain 00 and 11), (2) KK is closed under subtraction (a−b∈Ka - b \in K for all a,b∈Ka, b \in K), and (3) KK is closed under division (a⋅b−1∈Ka \cdot b^{-1} \in K for all a,b∈Ka, b \in K with b≠0b \neq 0). These conditions together guarantee that KK inherits the full field structure from FF. This is sometimes called the subfield test or subfield criterion.

Example

Problem: Show that Q\mathbb{Q} is a subfield of R\mathbb{R}.
Step 1: Check that Q\mathbb{Q} is nonempty and contains both identities.
0∈Q,1∈Q0 \in \mathbb{Q}, \quad 1 \in \mathbb{Q}
Step 2: Check closure under subtraction. If a=p1q1a = \frac{p_1}{q_1} and b=p2q2b = \frac{p_2}{q_2} are rational, then their difference is rational.
a−b=p1q2−p2q1q1q2∈Qa - b = \frac{p_1 q_2 - p_2 q_1}{q_1 q_2} \in \mathbb{Q}
Step 3: Check closure under division by nonzero elements. If b≠0b \neq 0, then b−1=q2p2b^{-1} = \frac{q_2}{p_2} is rational, so the product a⋅b−1a \cdot b^{-1} is rational.
a⋅b−1=p1q2q1p2∈Qa \cdot b^{-1} = \frac{p_1 q_2}{q_1 p_2} \in \mathbb{Q}
Answer: All three conditions of the subfield test are satisfied, so Q\mathbb{Q} is a subfield of R\mathbb{R}.

Why It Matters

Subfields appear throughout abstract algebra and number theory. Field extensions, built by studying how a larger field relates to a subfield, are the foundation of Galois theory and are used to prove results like the impossibility of trisecting an angle with compass and straightedge. Understanding subfields is also essential in coding theory and cryptography, where computations occur over finite fields and their subfields.

Common Mistakes

Mistake: Assuming every subring of a field is automatically a subfield.
Correction: A subring need not contain multiplicative inverses. For example, Z\mathbb{Z} is a subring of Q\mathbb{Q} but not a subfield, because integers like 22 have no multiplicative inverse in Z\mathbb{Z}. You must verify closure under division by nonzero elements.