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A highly efficient procedure for deciding if two given elements of      G L 2 ( Z [ −  1 ] ) {\text {G}}{{\text {L}}_2}(\mathbf {Z}[\sqrt { - 1} ])       are conjugate or not will be presented. It makes use of a continued fraction algorithm in      Z [ −  1 ] \mathbf {Z}[\sqrt { - 1} ]       and gives a fundamental unit of any given quadratic extension of      Q ( −  1 ) \mathbf {Q}(\sqrt { - 1} )      .

Conjugacy problem in 𝐺𝐿₂(𝑍[√-1]) and units of quadratic extensions of 𝑄(√-1)