By Wladyslaw Narkiewicz

The ebook offers with yes algebraic and arithmetical questions referring to polynomial mappings in a single or numerous variables. Algebraic homes of the hoop Int(R) of polynomials mapping a given ring R into itself are provided within the first half, beginning with classical result of Polya, Ostrowski and Skolem. the second one half bargains with absolutely invariant units of polynomial mappings F in a single or numerous variables, i.e. units X pleasing F(X)=X . This comprises particularly a learn of cyclic issues of such mappings on the subject of jewelry of algebrai integers. The textual content comprises numerous workouts and a listing of open difficulties.

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**Example text**

3. Put for i = 1 , 2 , . . , s: wi = k - vi, di = lrv', d = dl "" d,, Oi = d/di and define x 'fi(x) di If now t l , . 9 t8 are elements of R satisfying ~ tit9i = 1, i=1 then the polynomial $ Fk(X) = E tigi(X) i=1 is of degree k, maps R in R and its leading term equals 1/d, hence generates Ik. 2 (i)) that the sequence F0, F1 . . forms a regular basis. GERBOUD [89], who treated arbitrary Dedekind domains. Gm~BOUD [86], [885]). 3. ZANTEMA [82] showed that if the extension K / Q is normal with Galois group G, then the cohomology group H i ( G , U), where U = U ( K ) is the group of units of K, is related to the question whether K is a P61ya field.

I) The maximal kleals o f l n t ( Z ) are m a one-to-one correspondence with pairs ~, c], where p is a rational prime and c is a p-adic integer. This correspondence is given by [p,c] ~ M ( p , c ) = { f : f E I n t ( Z ) , f ( c ) E pZv}. (ii) Every non-maximal prime ideal of I n t ( Z ) is of the form P9 = g ( X ) Q [ X ] n I n t ( Z ) , where g E Z[X] is an irreducible polynomial. T w o ideals Pg and Ph coincide if and only if the polynomials g, h differ by a constant factor. 5. Let P be a non-zero prime ideal of Z[X], R = Z [ X ] / P and let p >_ O be the characteristics of the field of quotients of R.

Cf. Ruzs^ [71]. e. satisfies f ( m n ) = f ( , n ) f ( n ) for relatively prime m, n) equals either 0 or n k with a suit, able k. SOMAYAJULU [68]. PHONG [91]. STRAUS [52]. N6BAOER [76], who called a function f : R ---4 R compatible if for all ideals I of R the congruence a -- b (mod I) implies f(a) - f(b) (rood I). o. the structure of the semigroup (with regard to composition) of all compatible function in R = Z / n Z . This notion has been also regarded in greater generality in the theory of universal algebras.