- The algebraic closure of
in , is the set
-
(Fraleigh 31.12) Let
. Then -
A field is algebraically closed if every nonconstant polynomial in
has a zero in . -
(Fraleigh 31.15) A field
is algebraically closed if and only if every nonconstant polynomial in factors in into linear factors. -
Proof: In the forward case, let
be a polynomial. If is a zero of , then by algebraic closure and so is a factor of Conversely, if
is a factor of , then is a zero in which implies algebraic closure
-
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(Fraleigh 31.16) An algebraically closed field has no proper algebraic extensions
. -
(Fraleigh 31.17) Every field
has an algebraic closure, that is an algebraic extension that is algebraically closed. -
Intuition: An order relation exists for the set of all algebraic extension fields since they can contain one another.
Every chain in this set has an upper bound formed by taking the union of each element in the chain.
Zorn’s Lemma applies:
is the Maximal Element in the set of all algebraic extensions of . It can then be shown that is algebraically closed by maximality.
-
-
(Fraleigh 31.18) Fundamental Theorem of Algebra -
is an algebraically closed field. -
(Fraleigh 31.32) Every
that is not in the algebraic closure of over is transcendental over . -
(Fraleigh 31.35) No finite field of odd characteristic is algebraically closed.