Week 4-Lecture 17

Lecture 17 : Theorem of Kronecker

[tikarambhusal]

Why you have taken K[x] instead of K(x), give clearance

[ACh]

Let A be a K-algebra. Let there be an element x in A which is algebraic over K then we cannot say that the minimal polynomial of x over K is a prime polynomial in K[X] unless the algebra A is given to be an integral domain. For instance let us take K = IR and A = IR^2. Then A is a K-algebra. Let us take an element x = (1, 0) from IR^2. Then the minimal polynomial of x over K is X(X-1) which is clearly not a prime polynomial in K[X]. So we cannot go from (1) to (2) of the equivalence what you have mentioned in your lecture unless no further condition (e.g. integral domain etc.) has been imposed on the K-algebra A. Thank you very much.