In math, we normally proceed by learning elementary arithmetic and then elementary algebra. For me algebra is all about assigning a symbol to an unknown value and manipulating it to find its value. Now I don't understand how concepts like groups, fields, rings etc. fall into the algebra category. Wikipedia tells about the history of abstract algebra in brief: I've not studied general polynomial equations and higher degree forms of diophantine equations so I don't understand this. Can someone please explain me how concepts like groups, fields, rings etc. fit into my understanding of algebra which is basically assigning symbols to unknown and playing with it.