A divided by the first coefficient or SA. If is the coefficients may be read off directly. Thus the solutions in the diagram are −AX1/SA and −AX2/SA. The Carlyle circle, named after Thomas Carlyle, has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis. Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons. Generalization of quadratic equation The formula and its derivation remain correct if the coefficients , and are complex numbers, or more generally members of any field whose characteristic is not . (In a field of characteristic 2, the element is zero and it is impossible to divide by it.) The symbol in the formula should be understood as "either of the two elements whose square is , if such elements exist". In some fields, some elements have no square roots and some have two; only zero has just one square root, except in fields of characteristic . Even if a field does not contain a square root of some number, there is always a quadratic extension field which does, so the quadratic formula will always make sense as a formula in that extension field. Characteristic 2 In a field of characteristic , the quadratic formula, which relies on being a unit, does not hold. Consider the monic quadratic polynomial over a field of characteristic . If , then the solution reduces to extracting a square root, so the solution is and there is only one root since In summary, See quadratic residue for more information about extracting square roots in finite fields. In the case that , there are two distinct roots, but if the polynomial is irreducible, they cannot be expressed in terms of square roots of numbers in the coefficient field. Instead, define the 2-root of to be a root of the polynomial , an element of the splitting field of that polynomial. One verifies that is also a root. In terms of the 2-root operation, the two roots of the (non-monic) quadratic are and For example, let denote a multiplicative generator of the group of units of , the Galois field of order four (thus and are roots of over . Because , is the unique solution of the quadratic equation . On the other hand, the polynomial is irreducible over , but it splits over ,