How is the degree of the minimal polynomial related to the degree of a field extension?

field-theory galois-theory extension-field

$d=[L:K]$ is the dimension of $L$ when viewed as a vector space over $K$.

Consider $1,a,a^2,\dots,a^{d}$. This is a list of $d+1$ elements of $L$. Can they be linearly independent over $K$? What does it mean if they aren't?

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