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In the same way, an extension K 2 of K 1 can be constructed, etc. The union of all these extensions is the algebraic closure of K , because any polynomial with coefficients in this new field has its coefficients in some K n with sufficiently large n , and then its roots are in K n +1 , and hence in the union itself.
Because (a + 1) 2 = a, a + 1 is the unique solution of the quadratic equation x 2 + a = 0. On the other hand, the polynomial x 2 + ax + 1 is irreducible over F 4, but it splits over F 16, where it has the two roots ab and ab + a, where b is a root of x 2 + x + a in F 16. This is a special case of Artin–Schreier theory.
[4] [5] [6] Cramer's rule, implemented in a naive way, is computationally inefficient for systems of more than two or three equations. [ 7 ] In the case of n equations in n unknowns, it requires computation of n + 1 determinants, while Gaussian elimination produces the result with the same (up to a constant factor independent of n ...