Abstract Algebra Theory and Applications - Computer Science ...

NOTATION 389Symbol Description PageG/N factor group of G mod N 153ker φ kernel of φ 157G ′ commutator subgroup of G 168(a ij ) matrix 171O(n) orthogonal group 174‖x‖ length of a vector x 175SO(n) special orthogonal group 178E(n) Euclidean group 178O x orbit of x 205X g fixed point set of g 205G x isotropy subgroup of x 205X G set of fixed points in a G-set X 207N(H) normalizer of a subgroup H 222H the ring of quaternions 234char R characteristic of a ring R 238Z[i] the Gaussian integers 237Z (p) ring of integers localized at p 254R[x] ring of polynomials over R 257deg p(x) degree of p(x) 257R[x 1 , x 2 , . . . , x n ] ring of polynomials in n variables 260φ α evaluation homomorphism at α 260Q(x) field of rational functions over Q 281ν(a) Euclidean valuation of a 286F (x) field of rational functions in x 291F (x 1 , . . . , x n ) field of rational functions in x 1 , . . . , x n 291a ≼ b a is less than b 295a ∧ b meet of a and b 297a ∨ b join of a and b 297I largest element in a lattice 299O smallest element in a lattice 299a ′ complement of a in a lattice 299dim V dimension of a vector space V 317U ⊕ V direct sum of vector spaces U and V 320Hom(V, W ) set of all linear transformations from U to V 320V ∗ dual of a vector space V 320F (α 1 , . . . , α n ) smallest field containing F and α 1 , . . . , α n 325