Chapter 4: Discriminants

The Story…

You may have noticed the appearance of the discriminant  under the radical in the previous section where we discussed Cardano's method. Strangely, this number must be less than or equal to 0 if the cubic equation is to have only real roots. We begin by asking for a criterion in terms of cubic numbers for its associated polynomial to have only real roots. As it happens, that criterion is easy to state and prove.

Observation: If  is a cubic number then the characteristic polynomial dt( x - w ) has only real zeros if and only if  and  are all real.

Proof: This is simply because DT( x - w ) has the factorization:

But what does this mean in terms of cubic number w ?

Observation:  A cubic number  has  and  all real if and only if w is Hermitian, that is, if and only if w is equal to its conjugate:

Proof:  Recall what this means:

or,

Obviously, if , then  and  since, for example,

Thus, if , then  and  are all real.

Suppose, on the other hand that  and  are all real Recall the Vandermonde matrix

and let the column vector  be denoted A, and the column vector  be denoted W

Now notice again that: , and so, if the bar denotes matrix conjugation,  .  And since W is real, we have  . But this means

so , and  and it follows easily that w is Hermitian

We now have a simple criterion for a complex polynomial to have all roots real. It should be the characteristic polynomial of an Hermitian cubic number. Of course, it will have all roots real only if its coefficients are all real. So let us return to the simplified case of a "traceless" cubic polynomial (with real coefficients) of the form:  and ask under what circumstances will this be the characteristic polynomial of traceless, Hermitian cubic number w of the form:

Notice that for traceless, Hermitian cubic numbers, there is only one parameter, the complex number a. The condition

boils down to the statements:

Here,  is the “real part” of  that is

The polynomial p has all roots real if and only if a complex number a can be found satisfying these two conditions. We finish this section with the promised derivation of the discriminant relation.

Observation: A complex number a satisfying conditions:

can be found if and only if

Proof:  If  choose complex a so that  since of course   Then .  It follows from our assumption that  so we may further restrict a so that .

Thus the complex number a satisfies both conditions. On the other hand, suppose that a can be found satisfying both conditions. Then

and

and so .  It follows that .

We see that in the cubic case, the equation  (with d and e real) has all zeros real if and only if

. In this case, there is a complex number a (determined up to its conjugate) which determines a cubic number whose characteristic polynomial is . That cubic number is . From this it follows easily that the zeros of the polynomial have the form:

where  is a primitive cube root of unity.

We illustrate an interesting consequence of this in the exercise on the last page: Real Roots of Cubic Equations.