By H. Lausch, W. Nobauer

ISBN-10: 0720424550

ISBN-13: 9780720424553

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**Additional info for Algebra of Polynomials**

**Example text**

We will consider just algebras A such that I A 1 # 1, for j A j = 1 implies that A is polynomially complete for all varieties. These algebras will be called non-trivial algebras. 2. 4. , as we know, a finite field. Conversely, let Q be a finite field of order q. Then the number of distinct polynomial functions of the form a,- g;-I + a,-,5:-2 + ... , , q-1, is qq. For, if any two such functions are equal, we get a polynomial function ug-l&-lf . . ful(l+uo, by taking their difference which is the constant function with value 0.

Ck)fixing A elementwise and mapping b, to c,, i 1, . , k . A greatest (S)-root extension of A is an (S)-root extension A(c,, . , ck) of A such that, for any (S)-root extension A(b,, . , bk) of A, there exists a homomorphism from A(c,, . , cx) to A(b,, . , bk) fixing A elementwise and mapping c, to b,, i = 1, . , k . Clearly such a homomorphism is the only one with this property and is an epirnorphisrn. 41. Lemma. Every sohiable algebraic system ( S ) over ( A , %) it7 X = {x,, . , x k } has a greatest (5')-root extension, and any two greatest (S)-root extensioris of A are equivalent.

9,) = v,(aj, g,, . , gn), for some word wl. Let x E Fz ( A ) be defined by v, for u z b x(u,v> = c, for u = b. 23. We complete the proof of the theorem for case a). By definition of P,(A), every element of A is in P,(A). 22, every function of F,(A) which differs from a constant function only for one (gl,. ,g,) E A" is in P,(A). By repeating this argument, every function of F,(A) which differs from a constant function only for finitely many (gl, . ,g,) E A" is in P,(A). Since A is finite, A" is also finite, hence F,(A) = P,(A).