Exercise · ZW-CHARACTERISTIC-OF-A-RING-005 · ★☆☆ Basic
What is the characteristic of the field with 4 elements \(\mathbb{F}_4=\mathbb{F}_2[x]/(x^2+x+1)\)?
Choose one option.
Show answer
- C. \(2\)
Explanation
Its coefficients lie in \(\mathbb{Z}_2\), so \(1+1=0\) and the characteristic is 2, not 4 (the number of elements).
Review the concept: Characteristic of a ring
The characteristic of a ring \(R\) with identity \(1\) is the least positive integer \(n\) such that adding the identity to itself \(n\) times gives zero; if none exists, it is 0.
\[\operatorname{char}(R)=\min\{\,n>0 : n\cdot1=0\,\}\]
More practice problems: characteristic of a ring
- ZW-CHARACTERISTIC-OF-A-RING-001 What is the characteristic of ℤ_12?
- ZW-CHARACTERISTIC-OF-A-RING-002 What is the characteristic of ℤ_5×ℤ_7?
- ZW-CHARACTERISTIC-OF-A-RING-003 What is the characteristic of ℤ_4×ℤ_6?
- ZW-CHARACTERISTIC-OF-A-RING-004 What is the characteristic of the ring ℤ[x] of polynomials with integer coefficients?
- ZW-CHARACTERISTIC-OF-A-RING-006 In ℤ_5, which element equals 23·1?
- ZW-CHARACTERISTIC-OF-A-RING-007 Which of these numbers can be the characteristic of a finite integral domain?
- ZW-CHARACTERISTIC-OF-A-RING-008 What is the characteristic of the matrix ring M_2(ℤ_3)?
- ZW-CHARACTERISTIC-OF-A-RING-009 If R has identity and characteristic n>0, what is n· a for any a∈ R?
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