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**1. Homework Statement**

Show that the quaternion division ring H has infinitely many u satisfying u[tex]^{2}[/tex]=-1

**2. Homework Equations**

Elements of H is of the form a.1 +bi+cj+dk where a, b, c, d in [tex]\textsl{R}[/tex] ( reals) and i[tex]^{2}[/tex]= j[tex]^{2}[/tex]= k[tex]^{2}[/tex]=ijk = -1.

**3. The Attempt at a Solution**

Let u = a.1 +bi+cj+dk then u[tex]^{2}[/tex]=a[tex]^{2}[/tex]-b[tex]^{2}[/tex]-c[tex]^{2}[/tex]-d[tex]^{2}[/tex]+2a(bi+cj+dk)+2bicj+2bidk+2cjdk and this = -1 provided a=0 and -b[tex]^{2}[/tex]-c[tex]^{2}[/tex]-d[tex]^{2}[/tex]=1 and 2bicj+2bidk+2cjdk= 0 but I do not see how 2bicj+2bidk+2cjdk= 0.