Where do we use the theory of Discriminant of a number field.
I see that norm is used in determining the factorisation of an element in case of algebraic integers. But, I do not see use of Discriminant in solving similar problems.
Any reference would be appreciated.
One of the most significant invariant of an algebraic number field is the discriminant. One is tempted to say, apart from the degree, the most fundamental invariant. In simplest case of a quadratic equation, the discriminant tells us the behavior of solution, and of course, even its square roots gives us the solutions. To some extent the same is true for cubic equations, and the higher the degree of an equation less the influence of the discriminant is, but it always plays an important role.
For a field extension we shall define a discriminant for any basis. Of course this depends on the basis, but in a very perspicuous way. In the case of a number field, the ring of algebraic integers $A$ is a free module over $\mathbb Z$, and if the basises are confined to be $\mathbb Z$-basis for $A$, the discriminant is an integer independent of the the basis, is an invariant of the number field.
The discriminant serves several purposes at. Its main main feature is that it tells us in which primes a number fields ramify, or more generally, in which prime ideals an extension ramifies. Additionally the discriminant is a valuable tool to find $\mathbb Z$-basis for the ring $A$ of algebraic integers in a number field. To describe $A$ is in general a difficult task, and the discriminant is some times helpful. In relative situation, the situation is somehow more complicated, and the discriminant is well-defined just as an ideal.