The object of this paper is the calculation of the simple direct summands of the group algebra Q[G] when G = SL(2,q). We assume the character table of G is available and make the computation from that information. There is a well known procedure for finding the number of simple components. For each irreducible complex character γ, one forms the sum γ + γτ+ … of all the algebraic conjugates of γ. The sums obtained this way correspond one-to-one with the simple components. The dimension over Q of a simple component is determined from the characters corresponding to it. Furthermore, it is a full matrix ring over a division ring. Aside from the information obtained from the character table, then, all that is needed is a knowledge of the division rings that occur. The main result of the paper identifies the division rings in the simple component corresponding to the irreducible characters of G.