Kronecker's theorem says that a field extension can be shown as say, F(a) represented as F[x]/minimalpoly(a).
Say, Q[$\sqrt{2}$]=Q[x]/$(x^{2}-2$)
And a well known example is Q[$\sqrt{2}+\sqrt{3}$]=Q[$\sqrt{2},\sqrt{3}$]
But the minimum polys I will get will be different for both.
Minpoly for Q[$\sqrt{2},\sqrt{3}$] = $x^{4}-4x^{2}+1$ and
Q[$\sqrt{2}+\sqrt{3}$] = $x^{4}-10x^{2}+1$
So,
Is it that there are multiple generators for the same field extension.
How the minimum poly is different from ideals?
How this representation help more?
Take an even simpler example. As fields, $$\mathbb Q[\sqrt 2]= \mathbb Q[\sqrt 2+1]=\mathbb Q[a\sqrt2+b]$$ for any rational numbers $a,b$.
The point is, given a choice of generator $\alpha$ of a field $K$ over $\mathbb Q$ (so $K=\mathbb Q[\alpha]$), we can find the minimal polynomial $f(X)$ of $\alpha$ over $\mathbb Q$ and write $$K=\mathbb Q[\alpha]\cong \mathbb Q[X]/(f(X)).$$ However, the choice of polynomial depends entirely on the choice of $\alpha$, and this choice is far from unique.
This is analogous to choosing a basis of a vector space. Given an abstract vector space $V$ over a field $k$, we can choose a basis of $V$, which is the same as specifying an isomorphism $V\cong k^{\dim V}$.