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In American schools we what we call 'algebra' is just doing basic symbolic computation, like solving very simple equations.

The kind of algebra you're describing was called 'abstract algebra' at my university, and was an advanced pair of courses for math majors. The first semester covered group theory and the second semester covered ring theory.

Are German schools behind? In North America, learning algebra means we learn to solve for x/y : solving quadratic equations is pretty much the ceiling for "algebra" and even that isn't taught until grade 8, usually grade 9.

Im really not sure how though.

I'm from German too. We use algebra as the generic term for math more or less.

I wouldn't have assumed algebra would only mean fields and rings.

The Wikipedia article algebra also links to algebra over a field.