This is the problem:
Suppose that $K$ is an infinite field and $E|K$ is an extension with degree $n>1$. Prove that the quotient group $E^*/K^*$ is infinite.
i assume that the quotient group $E^*|K^*$ is finite, for example it's elements are $(b_1)k^*,...,(b_m)k^*$
also we can assume $E=K(a_1,a_2,...,a_n)$,
from linear algebra we know : $E|K$ can't be written as the finite union of it's proper subspaces,if we consider it as a vector space on $K$.i want to reach a contradiction...
any hint is welcomed.
If $A$ is a set of coset representatives of $K^\ast$, then $E^\ast = \bigcup_{a\in A} aK^\ast$. Furthermore, $aK^\ast \cup \{0\}$ is the span of $a$ if we view $E$ as a vector space over $K$. Thus, we have that
$$ \bigcup_{a\in A} \operatorname{span}\{a\} = \bigcup_{a \in A} (aK^\ast \cup \{0\}) = E^\ast \cup \{0\} = E. $$
Because $E \neq K$, $\operatorname{span}\{a\}$ must be a proper subspace of $E$. Because a vector space over an infinite field is not a finite union of proper subspaces, $A$ cannot be finite. Thus there is no finite set of coset representatives of $K^\ast$, so $E^\ast/K^\ast$ has infinite order.