Discriminant
From Academic Kids

In mathematics, a polynomial P(T) has a discriminant, which is a polynomial function of its coefficients, and discriminates the case of a multiple root (for which the graph of P(x) would touch the xaxis). This generalises to polynomials of any degree the case of a quadratic polynomial ax^{2} + bx + c, for which the discriminant is b^{2} − 4ac, the quantity under the square root sign in the formula for the roots (or quadratic formula). Discriminants in algebraic number theory are closely related, and contain information about ramification. In fact the more geometric types of ramification are also related to more abstract types of discriminant, making this a central algebraic idea in many applications.
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Discriminant of a polynomial
The discriminant of a polynomial is a number which can be easily computed from the coefficients of the polynomial and which is zero if and only if the polynomial has a multiple root. For instance, the discriminant of the polynomial ax^{2} + bx + c is b^{2} − 4ac.
For the general definition, suppose
 p(x) = x^{n} + a_{n−1}x^{n−1} + ... + a_{1}x + a_{0}
is a polynomial with real coefficients. The discriminant of this polynomial is defined as the determinant of the (2n − 1)×(2n − 1) matrix
1 a_{n−1} a_{n−2} . . . a_{0} 0 . . . 0 0 1 a_{n−1} a_{n−2} . . . a_{0} 0 . . 0 0 0 1 a_{n−1} a_{n−2} . . . a_{0} 0 . 0 . . . . . . . . . . . . . . 0 0 0 0 0 1 a_{n−1} a_{n−2} . . . a_{0} n (n−1)a_{n−1} (n2)a_{n−2} . . 1a_{1} 0 0 . . . 0 0 n (n−1)a_{n−1} (n−2)a_{n−2} . . 1a_{1} 0 0 . . 0 0 0 n (n−1)a_{n−1} (n−2)a_{n−2} . . 1a_{1} 0 0 . 0 . . . . . . . . . . . . . . 0 0 0 0 0 n (n−1)a_{n−1} a_{n−2} . . 1a_{1} 0 0 0 0 0 0 0 n (n−1)a_{n−1} a_{n−2} . . 1a_{1}
In the case n = 4, this discriminant looks like this:
<math>\begin{vmatrix}
& 1 & a_3 & a_2 & a_1 & a_0 & 0 & 0 \\ & 0 & 1 & a_3 & a_2 & a_1 & a_0 & 0 \\ & 0 & 0 & 1 & a_3 & a_2 & a_1 & a_0 \\ & 4 & 3a_3 & 2a_2 & 1a_1 & 0 & 0 & 0 \\ & 0 & 4 & 3a_3 & 2a_2 & 1a_1 & 0 & 0 \\ & 0 & 0 & 4 & 3a_3 & 2a_2 & 1a_1& 0 \\ & 0 & 0 & 0 & 4 & 3a_3 & 2a_2 & 1a_1 \\
\end{vmatrix}<math>
The discriminant of p(x) is thus equal to the resultant of p(x) and p'(x).
One can show that, up to sign, the discriminant is equal to
 Π_{i < j} (r_{i} − r_{j})^{2}
where r_{1}, ..., r_{n} are the (complex) numbers such that
 p(x) = (x − r_{1}) (x − r_{2}) ... (x − r_{n})
Therefore, p has a multiple root if and only if the discriminant is zero. Note however that this multiple root can be complex.
In order to compute discriminants, one does not evaluate the above determinant each time for different coefficient, but instead one evaluates it only once for general coefficients to get an easytouse formula. For instance, the discriminant of a polynomial of third degree is
 a_{1}^{2}a_{2}^{2} − 4a_{0}a_{2}^{3} − 4a_{1}^{3} + 18 a_{0}a_{1}a_{2} − 27a_{0}^{2}.
The discriminant can be defined for polynomials over arbitrary fields, in exactly the same fashion as above. The product formula involving the roots r_{i} remains valid; the roots have to be taken in some splitting field of the polynomial.
Discriminant of a conic
For conic section defined by real polynomials of the form
 Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F= 0,
the discriminant is equal to
 B^{2} − 4AC,
and determines the shape of the conic section. If the discriminant is less than 0, the equation is of an ellipse or a circle. If the discriminant equals 0, the equation is that of a parabola. If the discriminant is greater than 0, the equation is that of a hyperbola. This formula will not work for degenerate cases (when the polynomial factorises).
Case of a quadratic form
There is a substantive generalisation, to quadratic forms Q over any field K of characteristic ≠ 2. These can be written as a sum of terms
 a_{i}L_{i}^{2}
where the L_{i} are linear forms and 1 ≤ i ≤ n where n is the number of variables. Then the discriminant is the product of the a_{i}, taken in K/K^{2}, and is then welldefined (i.e., up to squares). A more invariant way to say this is as (the class of) the determinant of a symmetric matrix for Q.
Discriminant of an algebraic number field
If K is an algebraic number field and R its ring of integers, the discriminant of K is associated to R and in some sense measures how large R is. In the special case of R = Z[α] for some algebraic integer α in K, it is simple to define, as the discriminant of the minimal polynomial P_{α} of α. This suffices, for example, in the case of the Gaussian integers: we take P(T) = T^{2} + 1 for the choice α = i and calculate the discriminant as −4.
This in fact works for any quadratic field or cyclotomic field; but certainly not in general. There we can only be sure that Z[α] can be chosen, of finite index in R as abelian group. This gives a factor (of the index) which is awkward to apply. The correct definition comes through a recognition that the discriminant of a polynomial is a square of a Vandermonde determinant, and that determinant is what we should generalise. The analogue in the general case is this: let the ω_{i} be an integral basis (i.e. basis for R as Zmodule) and form
 det(ω_{i}^{(j)})
where the superscripts mean that we take the conjugates. This (squared) leads to the correct general definition.
Why this is the correct approach is best studied in terms of the real vector space
 <math>K \otimes \mathcal R\,<math>
and the embedding into it of R as a lattice. The determinant involved in the discriminant then has a simple interpretation as a volume of a fundamental region for R.
There is also a formula for the discriminant related to the quadratic form definition above, starting from the field trace. In the theory of Pontryagin duality for completions of K as local fields, the related different ideal occurs naturally in matching up Haar measures. This accounts for the role of the discriminant in the functional equation for the Dedekind zeta function, and thence in the analytic class number formula, and BrauerSiegel theorem.
A theorem of Stickelberger states that the discriminant D of an algebraic number field must be congruent to 0 or 1 modulo 4. A result of Kronecker is that D = 1 is possible only for the rational number field; this entails that every other number field has some ramified prime p in it. Lower bounds for discriminants, in terms of the degree, are proved by methods from the geometry of numbers and analytic number theory. In the case of an abelian extension, one of the results of class field theory (conductordiscriminant formula) is a factorisation of the discriminant according to characters, which in the case of an abelian extension of Q are Dirichlet characters.