Exercise · ZW-UNIT-OF-A-RING-010 · ★★★ Advanced
How many units \(u\) of \(\mathbb{Z}_8\) satisfy \(u^2=1\)?
Choose one option.
Show answer
- C. 4
Explanation
The units of \(\mathbb{Z}_8\) are \(1,3,5,7\), and their squares are \(1,9,25,49\), all congruent to \(1\) modulo \(8\). So all four satisfy \(u^2=1\), unlike a cyclic group, where there would be at most two solutions.
Review the concept: Unit of a ring
A unit of a ring \(R\) with identity \(1\) is an element \(u\) that has a multiplicative inverse \(v\) within \(R\).
\[uv=vu=1\]
More practice problems: unit of a ring
- ZW-UNIT-OF-A-RING-001 How many units does the ring ℤ_12 have?
- ZW-UNIT-OF-A-RING-002 Which of these elements is a unit of ℤ_10?
- ZW-UNIT-OF-A-RING-003 What is the multiplicative inverse of 5 in ℤ_17?
- ZW-UNIT-OF-A-RING-004 What are the units of the ring ℤ of integers?
- ZW-UNIT-OF-A-RING-005 How many units does the ring ℤ[i]=\a+bi: a,b∈ℤ\ have?
- ZW-UNIT-OF-A-RING-006 Which of these elements is a unit of ℤ[√2]?
- ZW-UNIT-OF-A-RING-007 If u and v are units of a commutative ring with identity, which of these elements is also a unit?
- ZW-UNIT-OF-A-RING-008 In a ring with 1ne0, can the element 0 be a unit?