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

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

:G₁

G₂ such that phi

(a · b) =

(a) ·

(b) (a,b

G₁). A function f : A maps

B 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 : A maps

B 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 :G maps

H 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:G maps

H is a homomorphism. The kernel of f is ker(f) = {x element of

G | f(x) = 1_H}. Let G₁ and G₂ be groups. An isomorphism from G₁ to G₂ is an injection phi

:G₁

G₂ such that phi

(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 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 :R maps

S 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

(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

R^x 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

R^x 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 element of

R. We say that x and y are associates if x = epsilon

y 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

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 {e_i} of V (finite or infinite) with the property that every element v of V can be uniquely written in the form v = sum over i

_ie_i. 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 dim_FV, 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 dim_FV to be infinity

Let

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 Irr_F( alpha

,X) The subfield F( alpha

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 A not equal

R. 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,...,f_m}, f_j 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 : G maps

H 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.