Let H be a subgroup of G. If for every a element of 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 element of I, r element of R, then a · r element of I and r · a element of 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 phi:G1mapsG2 such that phi(a · b) = phi(a) · phi(b)  (a,b element ofG1). A function f : AmapsB 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 : AmapsB is said to be surjective (or onto) if for every y element of B there exists x element of A such that f(x) = y. Let G and H be groups. A function f :GmapsH which satisfies f(g · g') = f(g) · f(g') for all g, g' element of G is called a homomorphism of G into H. If f:GmapsH is a homomorphism. The kernel of f is ker(f) = {x element of G | f(x) = 1H}. Let G1 and G2 be groups. An isomorphism from G1 to G2 is an injection phi:G1mapsG2 such that phi(a · b) = phi(a) · phi(b)  (a,b element ofG1 if phi is also surjective G1 is isomorphic to G2). Let H be a subgroup of G. If for every a element of 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 element of 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 element of G. For every x element of G, the exists y element of 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 :RmapsS such that for a,b element of 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. phi:nmapsphi(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 element of 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 element of Rx is not a unit of R, then x can be written as a product of irreducible elements of R.and (2) If x element of Rx is not a unit of R, and if x = pi1 ... pis = lambda1 ... lambdat are two expressions of x as a product of irreducible elements, then s = t and it is possible to renumber pi1 ... pis so that pi1 and lambda1 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 element of R. We say that x and y are associates if x = epsilony for some unit epsilon of R. If f has a leading coefficient 1, then we say that f is monic. We say that alpha is algebraic over F if there exists a nonzero polynomial f element of F[X] such that f(alpha) = 0. If alpha is not algebraic over F, then we say that alpha 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 = sum over ialphaiei. 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 infinity Let alpha element of E be algebraic over F. alpha the zero of a monic irreducible polynomial p element F[X]. The polynomial p is called the irreducible polynomial of alpha over F denoted IrrF(alpha,X) The subfield F(alpha1,...,alphan) 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 Anot equalR. 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 element of 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 : GmapsH is a homomorphism, then G/ker(f)isomorphic to 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.
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