Let be a field extension and pick any element . There is a morphism or more briefly , defined as the composition In other words, is a map of rings that does the obvious thing: it takes some polynomial and evaluates it at the element inside of .
Because is a principal ideal domain, the kernel of this map may be written as for some polynomial . We can further assume this polynomial is monic, since whether it is or not won’t affect the resulting ideal. We say that an element is algebraic over if, and only if, the polynomial is not zero. Otherwise, the element is said to be transcendental over . In other words, an element is transcendental if, and only if, the evaluation at that element is an injective ring homomorphism.
Usually, people define an algebraic element as an element such that there exists a non-zero polynomial which, when evaluated at that element, is zero; an element is then transcendental when it is not algebraic. It’s not hard to show that my definitions and the usual definitions are equivalent, so I’m not going to prove it here. The main reason for introducing my alternative definition is because it gives a context in which the minimal polynomial naturally appears as the kernel of an obvious map. Keep reading for the details.
For the rest of this article we suppose that is algebraic over , and thus the polynomial is non-zero (recall that we also supposed it is monic). This non-zero polynomial is called the minimal polynomial of . Because is (identified with) a subring of , and because is a field and thus an integral domain, the ring is also an integral domain. Hence, the ideal is prime, so the non-zero polynomial must be irreducible. Moreover, suppose for some other non-zero monic irreducible polynomial . In particular, , so there exists some polynomial such that . By irreducibility, is a unit, so and have the same degree, let’s call it . Equality of polynomials means they have the same coefficients at every degree; in particular, the coefficient at of is , while for it is . This forces , so . This justifies my usage of the word “the” when I defined the minimal polynomial.
I think this point of view on the minimal polynomial makes its properties clearer, and the importance of is better felt. For instance, with the “usual” definition, it’s not so hard to show that if is a polynomial such that , then divides . However, using my definition, this fact is immediate and needs almost no proof, since exactly means that is in the kernel of the evaluation map, which is .