alternative definition of algebraically closed
If (1) is true then we can prove by induction on degree of that every nonconstant polynomial splits completely over . Conversely, (2) (1) is trivial.
(2) (3) If is algebraic and , then is a root of a polynomial . By (2) splits over , which implies that . It follows that .
(3) (1) Let and a root of (in some extension of ). Then is an algebraic extension of , hence . ∎
Examples 1) The field of real numbers is not algebraically closed. Consider the equation . The square of a real number is always positive and cannot be so the equation has no roots.
2) The -adic field is not algebraically closed because the equation has no roots. Otherwise implies , which is false.
|Title||alternative definition of algebraically closed|
|Date of creation||2013-03-22 16:53:23|
|Last modified on||2013-03-22 16:53:23|
|Last modified by||polarbear (3475)|