What are the three conditions required for a nonempty set F to be called a field?

Conditions

• F is a nonempty set.
• The conditions must hold for all elements x, y, z in F.

Reasoning, step by step

1. Identify the first condition: F must form an additive abelian group.
2. Identify the second condition: The subset F - {0} must form an abelian group under multiplication.
3. Identify the third condition: Both left and right distributive laws of multiplication over addition must hold for all elements in F.
4. Conclude that satisfying these three conditions defines F as a field.

Example

The video explicitly lists the conditions on the handwritten page: '(i) F is an additive abelian Group.', '(ii) F-{0} is an abelian group w.r.t multiplication.', and '(iii) Left and right Distributive Laws hold in F.'

Common misconceptions

• Assuming that any nonempty set is automatically a field; it must satisfy specific axioms.
• Forgetting to exclude the zero element when defining the multiplicative group structure.

Watch the explanation

Explore next

• What five properties make F an additive abelian group?
• Why is zero excluded from the multiplicative group condition?
• What are the left and right distributive laws in a field?