The Stabilizer of the Power-free Points in ℚ ( 5 , i )

Let k ≥ 2 be an integer. A k-free point in a number field is an element of its ring of integers that is not divisible by the k th power of any prime ideal. We consider the biquadratic number field ℚ ( 5 , i ) . Through direct computation, we confirm the isomorphism between the stabilizer of its set of k -free points and the semidirect product of the unit group ℤ[τ, i ] × of its ring of integers with the Galois group Gal ( ℚ ( 5 , i ) / ℚ ) , where τ is the golden ratio 1 + 5 2 .