Algebraic Structures Exam 3

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Vector Space

Back 1

A set V is said to be a vector space over a field F if V is an Abelian group under addition (denoted by +) and, if for each a Î F and v Î V, there is an element av in V such that the following conditions hold for all a, b in F and all u, v in V.
1. a(v + u) = av + au
2. (a + b)v = av + bv
3. a(bv) = (ab)v
4. 1v = v

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Subspace

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Let V be a vector space over a field F and let U be a subset of V. We say that U is a subspace of V if U is also a vector space over F under the operations of V.

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Extension Field

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A field E is an extension field of a field F if F subset E and the operations of F are those of E restricted to F

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Fundamental Theorem of Field Theory (Kronecker's Theorem, 1887)

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Let F be a field and let f(x) be a nonconstant polynomial in F[x]. Then there is an extension field E of F in which f(x) has a zero

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Splitting Field

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Let E be an extension field of F and let f(x) in F[x]. We say that f(x) splits in E if f(x) can be factored as a product of linear factors in E[x]. We call E a splitting field for f(x) over F if f(x) splits in E but in no proper subfield of E.

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Existence of Splitting Fields

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Let F be a field and let f(x) be a nonconstant polynomial in F[x]. Then there exists a splitting field E for f(x) over F

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Theorem 20.3: F(a) » F[x]/Xp(x)\

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Let F be a field and let p(x) Î F[x] be an irreducible over F. If a is a zero of p(x) in some extension E of F, then F(a) is isomorphic to F[x]/Xp(x)\. Furthermore, if deg p(x) = n, then every member of F(a) can be uniquely expressed in the form
cn-1a n-1 + cn-2a n-2 + --- + c1a + c0 where c0, c1, ... , cn-1 Î F.

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F(a) » F(b)

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Let F be a field and let p(x) Î F[x] be irreducible over F. If a is a zero of p(x) in some extension E of F and b is a zero of p(x) in some extension of E' of F, then the fields F(a) and F(b) are isomorphic.

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Lemma

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Let F be a field, let p(x) Î F[x] be irreducible over F, and let a be a zero of p(x) in some extension of F. If f is a field isomorphism from F to F' and b is a zero of f(p(x)) in some extension of F', then there is an isomorphism from F(a) to F'(b) that agrees with f and F and carries a to b.

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Corollary: Splitting Fields Are Unique

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Let F be a field and let f(x) in F[x]. Then any two splitting fields of f(x) over F are isomorphic.

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Criterion for Multiple Zeros

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A polynomial f(x) over a field F has a multiple zero in some extension E if and only if f(x) and f''(x) have a common factor of positive degree in F[x].

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Zeros of an Irreducible

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Let f(x) be an irreducible polynomial over a field F. If F has characteristic 0, then f(x) has no multiple zeros. If F has characteristic p ¹ 0, then f(x) has a multiple zero only if it is of the form f(x) = gHx^p) for some g(x) in F[x].

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Types of Extensions

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Let E be an extension field of a field F and let a Î E. We call a algebraic over F if a is the zero of some nonzero polynomial in F[x]. If a is not algebraic over F, is called transcendental over F. An extension E of F is called an algebraic extension of F if every element of E is algebraic over F. If E is not an algebraic extension of F, it is called a transcendental extension of F. An extension of F of the form F(a) is called a simple extension of F.

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Characterization of Extensions

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Let E be an extension field of the field F and let a in E. If a is transcendental over F, then F(a) » F(x). If a is algebraic over F, then F(a) » F[x]/<p(x)>, where p(x) is a polynomial in F[x] of minimum degree such that p(a) = 0. Moreover, p(x) is irreducible over F

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Uniqueness Property

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If a is algebraic over a field F, then there is a unique monic irreducible polynomial p(x) in F[x] such that p(a) = 0.

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Divisibility Property

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Let a be algebraic over F, and let p(x) be the minimal polynomial for a over F. If f(x) in F[x] and f(a) = 0, then p(x) divides f(x) in F[x].

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Degree of an Extension

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Let E be an extension field of a field F. We say that E has degree n over F and write [E:F] = n if E has dimension n as a vector space over F. If [E:F] is finite, E is called a finite extension of F; otherwise, we say that E is an infinite extension of F

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Finite Implies Algebraic

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If E is a finite extension of F, then E is an algebraic extension of F.

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[K:F] = [K:E][E:F]

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Let K be a finite extension field of the field E and let E be a finite extension field of the field F. Then K is a finite extension field of F and [K:F] = [K:E][E:F].

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Primitive Element Theorem (Steinitz, 1910)

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If F is a field of characteristic 0, and a and b are algebraic over F, then there is an element c in F(a,b) such that F(a,b) = F(c).

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Primitive Element

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An element a with the property that E = F(a) is called a primitive element of E.

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Algebraic Closure

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For any extension E of a field F, the subfield of E of the elements that are algebraic over F is called the algebraic closure of F in E.

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Algebraically Closed

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A field that has no proper algebraic extension (and extension that is strictly greater than the field it is an extension of) is called algebraically closed.

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Classification of Finite Fields

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For each prime p and each positive integer n, there is, up to isomorphism, a unique finite field of order p^n

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Galois Field of order p^n

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For each field of prime-power p^n (there is only one field per prime-power), the field is unambiguously denoted by GF(p^n) and is called Galois field of order p^n
.

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Corollary 1

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[GF(p^n):GF(p)]=n

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GF(p^n) Contains an Element of Degree n

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Let a be a generator of the group of nonzero elements of GF(p^n) under multiplication. Then a is algebraic over
GF(p) of degree n

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Subfields of a Finite Field

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For each divisor m of n, GF(p^n) has a unique subfield of order p^n. Moreover, these are the only subfields of GF(p^n)

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Three ways to construct points

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Starting with points in a plane of F
1. Intersect two lines in F
2. Intersect a circle in F and a line in F
3. Intersect two circles in F

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Simple Groups

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A group is simple if its only normal subgroups are the identity subgroup and the group itself

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Sylow Test for Nonsimplicity

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Let n be a positive integer that is not prime, and let p be a prime divisor of n. If 1 is the only divisor of n that is
equal to 1 modulo p, then there does not exist a simple group of order n.