I have a basic question about the proof of "Every finite field extension is algebraic".
Given the extension $K\subset L$ with $n:=[L:K]$ and $a \in L$, the proof says, that we have a linearly dependent set $1,a,a^2, ...,a^{n+1}$, because their quantity $n+1$ is bigger than $n$, the dimension of the vectorspace of $L$ over $K$.
But why we can conclude this only by looking at the dimension? Otherwise why we can say, that $1,a,a^2,...,a^{n-1}$ would form a basis for the vectorspace $L$ over $K$. I mean for example given $a^2=a^3=...=a^{n+1}$, it wouln't be correct that $1,a,a^2,...,a^{n-1}$ is a linearly independent set. I hope you can help me. Thank you!
By definition, when a vector space has "dimension $n$", it means that any linearly independent subset has cardinality $n$ or less. Which means that if you have a set with $n+1$ elements, it is linearly dependent.
Having an arbitrary set of $n$ elements does not guarantee that you have a basis, because maybe they are linearly dependent; but that's not what your text says.