Abstract Algebra: An Introduction (3rd Edition)
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Let be a set of all matrices over the field of ... more
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Let be set of ordered pairs of rational numbers ... more
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The space of polynomial ring contains all ... more
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The space contains all polynomials of degree is ... more
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Let be set of tuples of vectors in . Let be a ... more
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Let be any set such that . It means every vector ... more
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Let . To show that the set spans over field ... more
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Let be subset of . Here we have to show linearly ... more
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Let subset be given. Then we have to show that ... more
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Let be a vector space over field . Let be a ... more
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Let be any vector space over field . Let be any ... more
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Let be a vector space over field . Let be a ... more
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We are given that the subset of vector space is ... more
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Let be a vector space over field . Let be a ... more
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Let be any set. To prove : is a basis of over .... more
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; Since is an extension field of field . ... more
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Let be a vector space over field . Let be any ... more
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Let be quotient space defined by where ... more
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Let be a vector space over field . Let, if ... more
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Since the set spans field over field . Therefore, ... more
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Let be the non zero real number. Suppose that is... more
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Let be set of polynomials and all polynomials in ... more
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Vector space denotes a set of all row vectors as... more
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Suppose , this shows that since . Thus over has... more
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The set will be basis if this set is linearly ... more
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Since the is basis for over , any vector of is ... more
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From , its enough to show that set forms a basis ... more
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Let be finitely generated vector space and be ... more
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Since is linearly independent subset of over , ... more
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For , if for some nonzero . Multiply both sides ... more
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Let is linearly independent over , this shows ... more
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S is contained in a basis of V ; Since V is a ... more
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Assume the contrary that [K:F] is infinite.Then ... more
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There is no field B such that . ; Assume the ... more
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is a subfield of K ; for all . Therefore ; is ... more
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; is the smallest field containing F and , ; ... more
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; F(a) is the smallest field containing F and a. ... more
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; Since , 3 and i must belong to themselves ; ... more
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is algebraic over F ; Since is algebraic over K... more
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c is algebraic over B ; Since is algebraic over A... more
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c is algebraic over ; Since is algebraic over F, ... more
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is algebraic over ; To be algebraic over , must ... more
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Let F and K be two fields such that . Let b be an... more
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6 ; Let ; Let . Clearly satisfies and f is the ... more
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; For , every element in must be the root of some... more
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Since F(u) is the smallest field containing F and ... more
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Let .f(x) is reducible over if and only if ... more
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For every element of to be algebraic over , it is... more
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It is enough to show that every element in ... more
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and ; Let .Square on both sides and simplify to ... more
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Since the minimal polynomial in is of prime ... more
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Since u is in and \F(u)\) is field, then is in... more
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For to be algebraic over , it is enough to show ... more
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Assume that .Since is the basis of over , there ... more
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Since , the basis of K is of the form where a is ... more
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Define by To prove , it is enough to prove that ... more
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Assume that in is algebraic over .Take an ... more
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Note that and . is the smallest field containing ... more
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Let SInce , the basis of must have n elements.... more
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{1,\sqrt{3},\sqrt{3}^2} ; x-2is the minimal ... more
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4 ; Q\subset Q (\sqrt{3})\subset Q(\sqrt{3})(i) = ... more
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4 ; x2-2 is a minimal polynomial of \sqrt{2} over ... more
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[K(u) : K] \leq [F(u) : F]. ; Let u be a root of... more
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[K(u) : F(u)] \leq [K : F] ; Every finite - ... more
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[F(u) : F] divides [K(u) : K] ; If K and F are ... more
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[K : F] is finite and K = F(u1,u2,...un) with ui ... more
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D is a field. ; Suppose that D \neq F and let d ... more
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Let be the set of all elements in which are ... more
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is an infinite dimensional algebraic extension of... more
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; Assume that Prove that Let and suppose that ... more
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are distinct positive integer. To prove that its ... more
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Splitting field of a polynomial splits into linear... more
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Consider the polynomial , now obtain the roots of ... more
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Splitting field of is . Since the minimal ... more
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Let . Then show that either or is a splitting ... more
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Let be a splitting field of a polynomial in ... more
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See that is the splitting field of over and ... more
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Since is algebraic over , there exists an ... more
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See that the irreducible polynomial in has roots... more
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Let be a finite field and , ... are arbitrary ... more
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See that the polynomial can be written as the ... more
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Let . See that . Factor to see that,. See that ... more
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. ; Let . Add -1 on both sides.Complete the ... more
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Let . Clearly, the coefficients of are in .From... more
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Let there be a root of which is irreducible ... more
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Let be an extension field of and be the subfields ... more
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Each irreducible polynomial has a degree one or is... more
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Suppose that and are not the same, then choose ... more
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Let the field be a normal extension. Now for a ... more
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Suppose is seprable over and is a field with Since... more
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It is given that is field extension of ... more
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Consider the map , where the field is of ... more
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Let .Using the concept of separability.Consider, ... more
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Let and be a positive integer.Let derivative of be... more
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; The field has characteristic 2 as in ... more
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Here, is the extension field of .Given is the ... more
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Let ,If and are relatively prime in then to prove ... more
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Here, is irreducible in .Given is separable, ... more
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The is contained in the set , hence .For elements... more
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Let is infinite and the be minimal polynomial of... more
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When and , from distributive law, the is , When... more
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Let be a identity of the . Therefore .Let is a ... more
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From claim its enough to show that the prime field... more
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Let be a characteristic of and characteristic of... more
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Consider is a finite field of order , that is . ... more
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Since is set of roots of , so if then, . This ... more
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Since all are belongs to and linearly ... more
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From , Let , use the binomial theorem,Each of the ... more
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Define a map given by , use binomial theorem,Each... more
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Prove the claim by considering the examples, first... more
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The field is finite filed with elements, then ... more
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From given claim its enough to show that every ... more
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It is given that the set of roots of , , is a ... more
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The both subfields of and of a finite field are... more
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Let be a given field with elenments. If then ... more