Let H be a subgroup of G. If for every a G, we have aHa-1 = H, then H is called a normal subgroup of G. Let R be a ring. An ideal of R is a subring I of R such that if a I, r R, then a · r I and r · a I. If H is a subgroup of G then aH = {a · h | a in G, h in H} is a left coset Ha = {h · a | a in G, h in H} is a right coset Let G1 and G2 be groups. An isomorphism from G1 to G2 is an injection :G1G2 such that (a · b) = (a) · (b)  (a,b G1). A function f : AB is said to be injective (or one to one) if whenever f(x) = f(y), we have x = y. A function that is both one-to-one and onto, that is if f(x) = f(y) then x = y, and for every y of the domain there is an x of the range so that f(x) = y. A function f : AB is said to be surjective (or onto) if for every y B there exists x A such that f(x) = y. Let G and H be groups. A function f :GH which satisfies f(g · g') = f(g) · f(g') for all g, g' G is called a homomorphism of G into H. If f:GH is a homomorphism. The kernel of f is ker(f) = {x G | f(x) = 1H}. Let G1 and G2 be groups. An isomorphism from G1 to G2 is an injection :G1G2 such that (a · b) = (a) · (b)  (a,b G1 if is also surjective G1 is isomorphic to G2). Let H be a subgroup of G. If for every a G, we have aHa-1 = H, then H is called a normal subgroup of G. The set of integers. commutative The set of rational numbers. The set of real numbers. The set of natural numbers. The complex numbers. The set residue classes mod n. The set of reduced residue classes mod n. For all a,b,c G, we have a·(b·c) = (a·b)·c. There exists an element e of G which has the property e · x = x · e = x for all x G. For every x G, the exists y G such that x · y = y · x = e. A commutative ring R is an integral domain if R contains no zero divisors. In other words, R is an integral domain if the product of any two nonzero elements of R is nonzero. Let R and S be rings. A ring homomorphism from R to S is a function f :RS such that for a,b R, we have (a) f(a + b) = f(a) + f(b),, (b) f(a · b) = f(a) · f(b). If <f is injective, f is a ring isomorphism A group is considered cyclic if it is generated by a single element. :n(n) = the number of integers a (1 < a < n) such that (a,n) = 1. If E is a field, then a subfield of E is a subset of E which is also a field with respect to the operations of E. Let R be a ring with identity 1, and let a R. If a has an inverse with respect to multiplication, then we say that a is a unit of R. We say that R is a unique factorization domain if the following two conditions are satisfied: (1) If x Rx is not a unit of R, then x can be written as a product of irreducible elements of R.and (2) If x Rx is not a unit of R, and if x = 1 ... s = 1 ... t are two expressions of x as a product of irreducible elements, then s = t and it is possible to renumber 1 ... s so that 1 and 1 are associates (1 < i < s). A ring R having the property that every ideal is principal is called a principal ideal ring (PIR). If, in addition, R is an integral domain, then R is called a principal ideal domain (PID). Let R be a ring x,y R. We say that x and y are associates if x = y for some unit of R. If f has a leading coefficient 1, then we say that f is monic. We say that is algebraic over F if there exists a nonzero polynomial f F[X] such that f() = 0. If is not algebraic over F, then we say that is transcendental over F. Let V be a vector space over F. A basis of V is a subset {ei} of V (finite or infinite) with the property that every element v of V can be uniquely written in the form v = iei. An ideal M of R is said to be a maximal ideal if (a) M is a proper ideal and (b) if A is a proper ideal containing M, then M = A. Let V be a vector space. If V has a finite set of generators, then we say that V is finite-dimensional and we define its dimension, denoted dimFV, to be the minimum possible number of elements in such a set. If V does not have a finite set of generators, then we say the V is infinite-dimensional, and we define dimFV to be Let E be algebraic over F. the zero of a monic irreducible polynomial p F[X]. The polynomial p is called the irreducible polynomial of over F denoted IrrF(,X) The subfield F(1,...,n) of E gotten by adjoining all zeros of f(X) to F is called the splitting field of f. An ideal A of R is said to be proper if AR. Let F be a field. A normal extension of F is an extension of F obtained by adjoining all zeros of a finite set of polynomials {f1,...,fm}, fj F[X]. An F-automorphism of E is an F-isomorphism of E onto itself. The set of all F-automorphisms of E will be denoted Gal(E/F). Gal(E/F) is called the Galois group of the extension E over F If f : GH is a homomorphism, then G/ker(f) im(f) A group G of order n is said to be solvable if it has a composition series whose composition factors are cyclic of prime order.