Exercise · ZW-UNIT-OF-A-RING-005 · ★☆☆ Basic
How many units does the ring \(\mathbb{Z}[i]=\{a+bi: a,b\in\mathbb{Z}\}\) have?
Choose one option.
Show answer
- C. 4
Explanation
The norm \(N(a+bi)=a^2+b^2\) is multiplicative, and \(a+bi\) is a unit if and only if \(N=1\). The integer solutions of \(a^2+b^2=1\) are \(\pm1\) and \(\pm i\): four units.
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-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?
- ZW-UNIT-OF-A-RING-009 For which of these values of n is every nonzero element of ℤ_n a unit?