Direct Sum algebraic number I'm doing a course in algebraic number theory and I don't understand the direct sum notation. How can the direct sum of fields be equal to a field? I thought the direct sum is a Cartesian product, where if you take the direct sum of $n$ fields, then each element of the direct sum is an $n$-tuple? 
For example take a field $K$ and an element a which is algebraic over $K$ of degree $n$. Then $K(a)=\bigoplus_{i=1}^{n-1} K(a^i)$. 
I'm very confused. If somebody can assist me here that would be great! 

The direct sum (or rather the product) of fields is not even an integral domain. 
In your context, $K(\alpha)$ is the direct sum of the $K\alpha^i$s as a $K$-vector space.
