# fifth roots of unity with integer coefficients

Considering the commutative ring Gamma 5, fifth roots of unity, with integer coefficients what can we say about elements which will be units? Are all possible values in the ring units, or are there conditions?

• Do you mean $\mathbb Z[\zeta_5]$ where $\zeta_5$ is a primitive 5th root of unity? – lhf May 13 '17 at 0:54
• Yep, that's what I meant. – Daniel May 13 '17 at 0:58
• Not every element of $\Bbb{Z}[\zeta_5]$ is a unit. For example $1-\zeta_5$ has norm $5$, $(1-\zeta_5)(1-\zeta_5^2)(1-\zeta_5^3)(1-\zeta_5^4)=5$ But obviously $\zeta_5$ is a unit. Less obviously $\zeta_5+\zeta_5^4$ is a unit which is not of finite order. Those two should generate all though. – sharding4 May 13 '17 at 1:01
• Possible duplicate of math.stackexchange.com/questions/3185/… and math.stackexchange.com/questions/956298/…. – lhf May 13 '17 at 1:01
• I'm not sure if I understand. Would you be able to direct me to somewhere with more information, or a good proof? – Daniel May 14 '17 at 7:55