Fields and finite fields
Objective. Analyse field structure and finite examples.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A field is a commutative ring with identity in which every nonzero element has a multiplicative inverse, so division by any nonzero element is possible; a finite field with q elements exists if and only if q=pⁿ for a prime p and positive integer n, and is denoted 𝔽_q. All finite fields of the same order are isomorphic, and the multiplicative group of nonzero elements of any finite field is cyclic. For example, 𝔽₄ (4 elements) is not ℤ/4ℤ, since ℤ/4ℤ has zero divisors (2×2=0 but 2≠0) and so is not a field; instead 𝔽₄ is built as 𝔽₂[x]/(x²+x+1).
Worked example
Which of ℤ/6ℤ and ℤ/7ℤ is a field?
- ℤ/6ℤ
- ℤ/7ℤ
- Both are fields
- Neither is a field
- 1A ring ℤ/nℤ is a field exactly when n is prime
- 2Check 6: it factors as 2×3, so it is not prime
- 3Check 7: it has no divisors other than 1 and itself, so it is prime
- 4Therefore ℤ/7ℤ is the field
Answer: ℤ/7ℤ
Concept mastery
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Practice
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Start practisingCommon mistake. Students assume ℤ/nℤ is a field for every n, but ℤ/4ℤ fails because 2 has no multiplicative inverse (2 times anything mod 4 is 0 or 2, never 1); ℤ/nℤ is a field exactly when n is prime.
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