Let R=F[x]/<x 2 -1>={a+bta,b in F; t 2 =1}. Show that a+bt is a unit in R if and only if a 2 b 2 . Solution If ab+bt is a unit there exists c,d in F such that (a+bt)(c+dt)=1 ac+bd=1 ad+bc=0 If we multiply the first equation by b we get abc+b^2d=b But bc=-ad from the second equation and we replace it in the term abc We get -a^2d+b^2d=b d(b^2-a^2)=b If a^2=b^2, then we obtain d*0=b, so b=0, so a^2=0, so a=0 because it is in a field Conversely, suppose a^2b^2 Then we choose d=b*(b^2-a^2) -1 and c=-ad*b -1 =-a*(b^2-a^2) -1 Then we have ac+bd=-a^2*(b^2-a^2) -1 +b^2*(b^2-a^2) -1 =1 and ad+bc=0 which shows (a+bt)(c+dt)=1 so a+bt is a unit .