By H. Lausch, W. Nobauer

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N,, a,). This establishes the theorem. 2. Let G be a group we will regard as an algebra of 23 and suppose X = {x,, . , xk}. We write G(X, 53) = G[x,, . , xk] and want to find a normal form system for this algebra. Again we first simplify our nota- 30 POLYNOMIALS AND POLYNOMIAL FUNCTIONS CH. 1 tion. Let M be the additive group of integers and M k the direct product of k copies of M . (0, 0, . ,0) E M k will be denoted by 0. An ordered pair (i,, . , i,), (Z, . , 1,) of elements of M k is called reducible if, for some index v, i, f 0, iv+l = iV+,= .

WnP1)w,. If w1 = . . = w, = w, we write wl... w,,= w", and set wo = 1 . Then ( a n y ) + . . (alx)+ao, a, E R, v = 0, . . , n, is a well. . +alx+a,. 11. Theorem. Let % betheset of all words a n y + . . +a,x+ao where n a 0, a,€ R , for t = 0, . , n, and a,, # 0. Then %U{O} is a normal form system of R [ x ] . Proof. 11. a) We have to show that, for every representation p = w(a,, x ) of an element p E R [ x ] ,we can find a word in % representing p , in a finite number of steps. We proceed by induction on the minimal rank r of w ( a , x).

Ug-l = 0, else the polynomial U , - ~ X ~ - ~ +. . u,x+ uo would have at most q-1 different roots, a contradiction. e. Q is 1-polynohially complete. The functions P,-~&) t~-l+p,&) 5;-,+ . . (&)E PI@), 2’ = 0, . , q - 1, are functions of P,(Q). (51) 5; c 0 = v=q-1 6;. r”(E1) § 12 SOME EXAMPLES OF POLYNOMIALLY COMPLETE ALGEBRAS Let a € Q be arbitrary, then 0 ,,=q-l p,(a) ti 0 = v=q-1 39 r,(a) ti, and by the argument from above we conclude p,(a) = rv(a), v = 0, . e. p,(5,) = r&), v = 0, . , q- 1.