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Abstract Algebra




                    Notes          Let F be an extension field of K. The dimension of F as a vector space over K is called the degree
                                   of F over K, denoted by [F : K]. If the dimension of F over K is finite, then F is said to be a finite
                                   extension of K. Let F  be an extension field of K and let u  F. The following conditions are
                                   equivalent: (1) u is algebraic over K; (2) K(u) is a finite extension of K; (3) u belongs to a finite
                                   extension of K.
                                   Never underestimate the power of counting: the next result is crucial. If we have a tower of
                                   extensions K  E  F, where E is finite over K and F is finite over E, then F is finite over K, and
                                   [F : K] = [F : E][E : K]. This has a useful corollary, which states that the degree of any element of
                                   F is a divisor of [F : K].
                                   Let K be a field and let f(x) = a  + a x + ... + a x  be a polynomial in K[x] of degree n > 0. An
                                                                          n
                                                                        n
                                                                1
                                                            0
                                   extension field F of K is called a splitting field for f(x) over K if there exist elements r , r ,..., r n
                                                                                                          2
                                                                                                        1
                                    F such that
                                   (i)  f(x) = a (x – r )(x – r ) ... (x – rn) and
                                                  1
                                             n
                                                       2
                                   (ii)  F = K(r , r ..., r ).
                                             1
                                               2,
                                                   n
                                   In this situation we usually say that f(x) splits over the field F. The elements r , r ,..., r  are roots
                                                                                                   2
                                                                                                       n
                                                                                                 1
                                   of f(x), and so F  is obtained by  adjoining to K a  complete set  of roots of f(x). An  induction
                                   argument (on the degree of f(x)) can be given to show that splitting fields always exist. Theorem
                                   states that if f(x)  K[x] is a polynomial of degree n > 0, then there exists a splitting field F for f(x)
                                   over K, with [F : K]  n!.
                                   The uniqueness of splitting fields follows from two lemmas. Let  : K  L be an isomorphism of
                                   fields. Let F be an extension field of K such that F = K(u) for an algebraic element u  F. Let p(x)
                                   be the minimal polynomial of u over K. If v is any root of the image q(x) of p(x) under , and
                                   E = L(v), then there is a unique way to extend  to an isomorphism  : F  E such that (u) = v and
                                   (a) = (a) for all a  K. The required isomorphism  : K(u)  L(v) must have the form
                                                    (a  + a u + ... + a u ) = (a ) + (a )v + ... + (a )v n-1
                                                                    n-1
                                                         1
                                                      0
                                                                          0
                                                                                1
                                                                 n-1
                                                                                         n-1
                                   The second lemma is stated as follows. Let F be a splitting field for the polynomial f(x)  K[x]. If
                                    : K  L is a field isomorphism that maps f(x) to g(x)  L[x] and E is a splitting field for g(x) over
                                   L, then there exists an isomorphism  : F  E such that (a) = (a) for all a  K. The proof uses
                                   induction on the degree of f(x), together with the previous lemma.
                                   Theorem states that the splitting field over the field K of a polynomial f(x)  K[x] is unique up
                                   to isomorphism. Among other things, this result has important consequences for finite fields.
                                   Self Assessment
                                   1.  If F is an extension field k and u  F is algebraic over K, then their exists a ...............
                                       (a)  different                (b)  finite
                                       (c)  infinite                 (d)  unique

                                   2.  The monic  polynomial P(x)  of minimal  degree in  K[x] such  that P(u)  = 0  is called  is
                                       ............... of r over K and its degree is called the degree of u over K.
                                       (a)  maximal  polynomial      (b)  minimal  polynomial

                                       (c)  finite  polynomial       (d)  infinite  polynomial
                                   3.  The dimension of F as a vector space K is called the ............... of F over K, denoted by [f : k]
                                       (a)  range                    (b)  domain
                                       (c)  degree                   (d)  field




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