I've been alerted of one possible mistake made in pblm 52 (I don't know
yet how many made it). I feel I should point this out to everybody in 
writing, and the grader is advised not to make deductions for this one,
due to the deplorably low presence of complex numbers in UG curricula:

Once you have factored 

      X^4+1  =  (X^2 + sqrt2 X + 1)(X^2 - sqrt2 X + 1)                (1)

over real coefficients, you can use the quadratic formula to factor the
quadratics further, admitting now complex coefficients. If you do this, you
are fine and encounter no problem.

However, you may immediately *see* a simple factorization into quadratics, 
once you allow complex coefficients, namely: 

      X^4+1  = (X^2-i)(X^2+i)                                         (2)

This is still fine so far, but then it seduces you to further factorization

      X^4+1  = (X + sqrt(i))(X - sqrt(i))(X + i sqrt(i))(X - i sqrt(i))
                                                          (---NOT GOOD---)

This you should *not* do: the reason is that square roots of complex numbers
pose difficulties which you may not be prepared for: I would have to ask you:
what is sqrt(i) ?  Say, written out in the form  a+bi  ?  Are you sure there
is a complex number that qualifies as sqrt(i) at all? These questions can 
be answered in some way, but *you* can probably not answer them appropriately, 
and you are not required to. 

Let me show you the key problem: When I ask you for sqrt9, you'll say 3, 
not -3. Because we all agree that among the two real numbers whose square
is 9, we designate the positive one as the square root. Since among complex 
numbers we cannot distinguish positive numbers, this option is not available 
any more, and sqrt(i) remains ambiguous. But an ambiguous expression is a 
meaningless expression in mathematics. 
So the upshot is: You should *never* write anything like sqrt(i) or 
sqrt(2+3i), unless/until you have covered a thorough course in complex 
variables. 

Here is the result you get from further factoring (1) using the 
quadratic formula:

           1 + i           1 - i          -1 + i          -1 - i   
    ( X - ------- ) ( X - ------- ) ( X - ------- ) ( X - ------- )    (**)
           sqrt2           sqrt2           sqrt2           sqrt2   

Depending on which pairs you combine to form quadratics, you retrieve
either (1) or (2) or a third factorization

      X^4+1 = (X^2 + i sqrt2 X - 1)(X^2 - i sqrt2 X - 1)                (3)

You could also have found factors from (**) by finding the monic gcd
of one factor from (1) and one factor from (2) using the euclidean algorithm:
For instance (try it)

   GCD ( X^2 + sqrt2 X + 1 , X^2+i ) = X - (-1+i)/sqrt2

This finding of GCD's is of course just the reverse procedure of combining
linear factors from (**) to get (1),(2),(3).