The usual practice in any discussion of an inner-product space is to restrict the field over which the inner-product space is defined to the field of complex numbers. In defining the inner-product function, (x,y), a second function is needed; namely the conjugate function (x,y)* so that (x,y) ± (y,x)*. We will attempt to generalize this concept by investigating the existence of a conjugate function defined on fields other than the field of complex numbers and relate this function to an inner-product function defined on a linear space L over these fields.