with observations of an x variable, and a y variable whose expected values lie
on a straight line, and whose residual errors are normally distributed with
unit variance, and are independent, how do I estimate the parameters of the
line?

I think I have a maximum likelihood estimate by choosing the line which
minimizes the sum of squared distances from the observations to the line, but
am not sure how to prove this.

I think this generalizes to multiple dimensions but also am not sure how to
prove this.

This seems like a sufficiently interesting problem so that I am certain there
must be prior publications which offer clarifications for the comments above.

Can you guide me to some reading, please.