F. Group Characters.
If G is a multiplicative group, F a field and G a homomorphism mapping G into F, then o is called a character of G in F. By homomor- phism is meant a mapping (j such that for a, B any two elements of G,
(If o(a) = 0 for one element a, then o(x) = 0 for each x ( G, since <r( ay) = c(a)'o(y) = 0 and ay takes all values in G when y assumes all values in G).
The characters olro2,. ■ ■ ,anare called dependent if there exist elements a a,, . . . , a„ not all zero in F such that pendence relation is called non-trivial. If the characters are not dependent they are called independent.
THEOREM 12. If G is a group and av o2,. . . , on are n mutually distinct characters of G in a field F, then a v U^ . . . , on are independent.
One character cannot be dependent, since a 1a1(x) - 0 implies a} = 0 due to the assumption that a.(x) ^ 0. Suppose n > 1.
We make the inductive assumption that no set of less than n distinct characters is dependent. Suppose now that is a non-trivial dependence between the a s. None of the elements is zero, else we should have a dependence between less than n characters contrary to our inductive assumption. Since o i and an are distinct, there exists an element a. in Gi SUCh that (7j (a) é on(a). Multiply the relation between the ff's b y a "We obtain a relation and since b, i o, we get a,( a) — G. (a) contrary to the choice of a. Thus, (* * ) is a non-trivial dependence between (j j, ff2, . » °n- 1 w^ich is contrary to our inductive assumption.
Corollary. If E and E 1 are two fields, and CTj , a2 , . . , a are n mutually distinct isomorphisms mapping E into E 1 , then al , . , on are independent. (Where "independent'' again means there exists no non-trivial dependence a ^o t (x ) + . + a a (x ) - 0 which holds for every x ( E).
Replace in this relation x by ax. We have
Subtracting the latter from (*) we have
The coefficient of <7j (x ) in this relation is not 0, otherwise we should
This follows from Theorem 12, since E without the 0 is a group
and the a's defined in this group are mutually distinct characters.
If t7p > • • • > O are isomorphisms of a field E into a field E 1 , then each element a of E such that <7j(a) = o,(a) = . . . = on ( a ) is called a fixed point of E under Oj , 02 , ■ ■ ■ , • This name is chosen because in the case where the (jys are automorphisms and is the identity, i.e., (x) = x, we have o\ (x) = x for a fixed point.
Lemma. The set of fixed points of E is a subfield of E. We shall call this subfield the fixed field.
Suppose to the contrary that
We shall show that
we are led to a contradiction.
3e a generating system of E Over F. In the homogeneous linear equations
For if a and b are fixed points, then
Thus, the sum and product of two fixed points is a fixed point, and the inverse of a fixed point is a fixed point. Clearly, the negative of a fixed point is a fixed point.
THEOREM 13. If ov. . . , 0"n are n mutually distinct isomorphisms of a field E into a field E1 » and if F is the fixed field of E, then
(E/F ) >_n .
there are more unknowns than equations SO that there exists a non- trivial solution which, we may suppose, x j,x 2, . . . ,xn denotes. For any element a in E we can find a,, aJ(. . . , a, in F such that a =. a jtjjj + . . . + a (d, We multiply the first equation by cr ( a )( the second by a. (a ), and so on. Using that a. f F, hence that
and also that
Corollary, If a,, a2, . . . , on are automorphisms of the field E, and F is the fixed field, then (E/F) > n.
Adding these last equations and using
This, however, is a non-trivial dependence relation between a a2, . . . , an which cannot exist according to the corollary of Theorem 12.
If F is a subfield of the field E, and <y an automorphism of E, we shall say that a leaves F fixed if for each element a of F, o(a) = a.
(t7(x) = y), then we shall call a'1 the mapping of y into x, i.e.,o-~l(y) = X the inverse of cr. The reader may readily verify that cr1 is an automorphism. The automorphism I (x ) = x shall be called the
Lemma. If E is an extension field of F, the set G of automorphisms which leave F fixed is a group.
The product of two automorphisms which leave F fixed clearly leaves F fixed. Also, the inverse of any automorphism in G is in G.
The reader will observe that G, the set of automorphisms which leave F fixed, does not necessarily have F as its fixed field. It may be that certain elements in E which do not belong to F are left fixed by every automorphism which leaves F fixed. Thus, the fixed field of G may be larger than F.