Exercise · ZW-UNIT-OF-A-RING-006 · ★★☆ Intermediate
Which of these elements is a unit of \(\mathbb{Z}[\sqrt2]\)?
Choose one option.
Show answer
- A. \(1+\sqrt2\)
Explanation
We have \((1+\sqrt2)(\sqrt2-1)=2-1=1\), and \(\sqrt2-1\) lies in the ring. The others have norm \(a^2-2b^2\) equal to \(2\), \(7\) and \(-4\), different from \(\pm1\), so they are not 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-005 How many units does the ring ℤ[i]=\a+bi: a,b∈ℤ\ have?
- 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?