K $\subset L $ and $a \in L$. Suppose that a set $\{1,a,a^2,a^3,a^4 \}$ is linear dependent over $K$. Check that $a$ is an algebraic over $K$.
I suppose that I have to find a polynomial $f$ with coefficients from $K$ such that $f(a)=0$. But I have no idea what to do with this linear dependent set. If it were linear independent it would be a basis of $K(a)$. But it is not.
Since the set $\lbrace 1,a,a^2,a^3,a^4\rbrace$ is linear dependent over $K$ there must be some coefficients $c_i$ in $K$ (NOT all of them are zero) with $$c_01+c_1a+c_2a^2+c_3a^3+c_4a^4=0$$ but now $a$ is a root of the polynomial $f\in K[x]$ which is given by $$f(x)=c_0+c_1x+c_2x^2+c_3x^3+c_4x^4$$