field4 http://www.openmath.org/cd/field4.ocd 2006-06-01 2004-06-01 1 1 experimental A CD of functions for morphisms of fields. Written by Arjeh M. Cohen 2004-07-07 automorphism_group This is a function with a single argument which must be a field. It refers to the automorphism group of its argument. homomorphism_by_generators This is a function with three arguments the first two of which must be fields F and K. The third argument should be a set or a list L of ordered pairs (lists of length 2). Each pair [x,y] from L consists of an element x from F and an element y from K. when applied to F, K, and L, the symbol represents the homomorphism from F to K that maps the first entry x of each pair [x,y] to the second entry y of the same pair. field_by_poly_map Same as quotient_by_poly_map in CD ring5, except that R and the quotient ring R[X]/(f) are fields (so f is irreducible in R[X]). An element aX + b of the finite field GF(3)[X]/(X^2+1) is represented by 3 1 10 12 0 1 field_by_poly_vector This symbol is a binary function. Its first argument should be a field_by_poly(R,f). Its second argument should be a list L of elements of F, the coefficient field of the univariate polynomial ring R = F[X]. The length of the list L should be equal to the degree d of f. When applied to R and L, it represents the element L[0] + L[1] x + L[2] x^2 + ... + L[d-1] ^(d-1) of R/(f), where x stands for the image of x under the natural quotient map R -> R/(f). If the first argument is a field_by_conway(p,n), defined in the CD finfield1, then the same interpretation holds, where R and f are respectively poly_ring_d(GFp(p),1) and conway_polynomial(p,n). later The element x+1 of the Conway field of order 4: 22 1 1