Abstract Algebra: An Introduction (3rd Edition)

Chapter 11

Verified Answer ✓

Let be a set of all matrices over the field of ... more

Verified Answer ✓

Let be set of ordered pairs of rational numbers ... more

Verified Answer ✓

The space of polynomial ring contains all ... more

Verified Answer ✓

The space contains all polynomials of degree is ... more

Verified Answer ✓

Let be set of tuples of vectors in . Let be a ... more

Verified Answer ✓

Let be any set such that . It means every vector ... more

Verified Answer ✓

Let . To show that the set spans over field ... more

Verified Answer ✓

Let be subset of . Here we have to show linearly ... more

Verified Answer ✓

Let subset be given. Then we have to show that ... more

Verified Answer ✓

Let be a vector space over field . Let be a ... more

Verified Answer ✓

Let be any vector space over field . Let be any ... more

Verified Answer ✓

Let be a vector space over field . Let be a ... more

Verified Answer ✓

We are given that the subset of vector space is ... more

Verified Answer ✓

Let be a vector space over field . Let be a ... more

Verified Answer ✓

Let be any set. To prove : is a basis of over .... more

Verified Answer ✓

; Since is an extension field of field . ... more

Verified Answer ✓

Let be a vector space over field . Let be any ... more

Verified Answer ✓

Let be quotient space defined by where ... more

Verified Answer ✓

Let be a vector space over field . Let, if ... more

Verified Answer ✓

Since the set spans field over field . Therefore, ... more

Verified Answer ✓

Let be the non zero real number. Suppose that is... more

Verified Answer ✓

Let be set of polynomials and all polynomials in ... more

Verified Answer ✓

Vector space denotes a set of all row vectors as... more

Verified Answer ✓

Suppose , this shows that since . Thus over has... more

Verified Answer ✓

The set will be basis if this set is linearly ... more

Verified Answer ✓

Since the is basis for over , any vector of is ... more

Verified Answer ✓

From , its enough to show that set forms a basis ... more

Verified Answer ✓

Let be finitely generated vector space and be ... more

Verified Answer ✓

Since  is linearly independent subset of over , ... more

Verified Answer ✓

For , if for some nonzero . Multiply both sides ... more

Verified Answer ✓

Let is linearly independent over , this shows ... more

Verified Answer ✓

S is contained in a basis of V ; Since V is a ... more

Verified Answer ✓

Assume the contrary that [K:F] is infinite.Then ... more

Verified Answer ✓

There is no field B such that . ; Assume the ... more

Verified Answer ✓

is a subfield of K ; for all . Therefore ; is ... more

Verified Answer ✓

; is the smallest field containing F and ,   ; ... more

Verified Answer ✓

; F(a) is the smallest field containing F and a. ... more

Verified Answer ✓

; Since , 3 and i must belong to themselves ; ... more

Verified Answer ✓

is algebraic over F ; Since is algebraic over K... more

Verified Answer ✓

c is algebraic over B ; Since is algebraic over A... more

Verified Answer ✓

c is algebraic over ; Since is algebraic over F, ... more

Verified Answer ✓

is algebraic over ; To be algebraic over , must ... more

Verified Answer ✓

Let F and K be two fields such that  . Let b be an... more

Verified Answer ✓

6 ; Let ; Let . Clearly satisfies and f is the ... more

Verified Answer ✓

; For , every element in must be the root of some... more

Verified Answer ✓

Since F(u) is the smallest field containing F and ... more

Verified Answer ✓

Let .f(x) is reducible over if and only if ... more

Verified Answer ✓

For every element of  to be algebraic over , it is... more

Verified Answer ✓

It is enough to show that every element in ... more

Verified Answer ✓

and ; Let .Square on both sides and simplify to ... more

Verified Answer ✓

Since the minimal polynomial in is of prime ... more

Verified Answer ✓

Since u is in and \F(u)\) is field, then is in... more

Verified Answer ✓

For to be algebraic over , it is enough to show ... more

Verified Answer ✓

Assume that .Since is the basis of over , there ... more

Verified Answer ✓

Since , the basis of K is of the form where a is ... more

Verified Answer ✓

Define by To prove , it is enough to prove that ... more

Verified Answer ✓

Assume that in is algebraic over .Take an ... more

Verified Answer ✓

Note that and . is the smallest field containing ... more

Verified Answer ✓

Let SInce , the basis of must have n elements.... more

Verified Answer ✓

{1,\sqrt{3},\sqrt{3}^2} ; x-2is the minimal ... more

Verified Answer ✓

4 ; Q\subset Q (\sqrt{3})\subset Q(\sqrt{3})(i) = ... more

Verified Answer ✓

4 ; x2-2 is a minimal polynomial of \sqrt{2} over ... more

Verified Answer ✓

[K(u) : K] \leq  [F(u) : F]. ;  Let u be a root of... more

Verified Answer ✓

[K(u) : F(u)] \leq [K : F]  ; Every finite - ... more

Verified Answer ✓

[F(u) : F] divides [K(u) : K]  ; If K and F are ... more

Verified Answer ✓

[K : F] is finite and  K = F(u1,u2,...un) with ui ... more

Verified Answer ✓

 D is a field. ; Suppose that D \neq F and let d ... more

Verified Answer ✓

Let be the set of all elements in which are ... more

Verified Answer ✓

is an infinite dimensional algebraic extension of... more

Verified Answer ✓

; Assume that Prove that  Let and suppose that ... more

Verified Answer ✓

are distinct positive integer. To prove that its ... more

Verified Answer ✓

Splitting field of a polynomial splits into linear... more

Verified Answer ✓

Consider the polynomial , now obtain the roots of ... more

Verified Answer ✓

Splitting field of is . Since the minimal ... more

Verified Answer ✓

Let . Then show that either or is a splitting ... more

Verified Answer ✓

Let be a splitting field of a polynomial in ... more

Verified Answer ✓

See that  is the splitting field of over and ... more

Verified Answer ✓

Since is algebraic over , there exists an ... more

Verified Answer ✓

See that the irreducible polynomial in has roots... more

Verified Answer ✓

Let be a finite field and , ... are arbitrary ... more

Verified Answer ✓

See that the polynomial can be written as the ... more

Verified Answer ✓

Let . See that . Factor to see that,. See that ... more

Verified Answer ✓

. ; Let . Add -1 on both sides.Complete the ... more

Verified Answer ✓

Let . Clearly, the coefficients of are in .From... more

Verified Answer ✓

Let there be a root of which is irreducible ... more

Verified Answer ✓

Let be an extension field of and be the subfields ... more

Verified Answer ✓

Each irreducible polynomial has a degree one or is... more

Verified Answer ✓

Suppose that and are not the same, then choose ... more

Verified Answer ✓

Let the field be a normal extension. Now for a ... more

Verified Answer ✓

Suppose is seprable over and is a field with Since... more

Verified Answer ✓

It is given that is field extension of ... more

Verified Answer ✓

Consider the map  , where the field is of ... more

Verified Answer ✓

Let .Using the concept of separability.Consider, ... more

Verified Answer ✓

Let and be a positive integer.Let derivative of be... more

Verified Answer ✓

; The field has characteristic  2 as in  ... more

Verified Answer ✓

Here, is the extension field of .Given is the ... more

Verified Answer ✓

Let ,If and are relatively prime in then to prove ... more

Verified Answer ✓

Here, is irreducible in .Given is separable, ... more

Verified Answer ✓

The is contained in the set , hence .For elements... more

Verified Answer ✓

Let is infinite and the be minimal polynomial of... more

Verified Answer ✓

When and , from distributive law, the is , When... more

Verified Answer ✓

Let be a identity of the . Therefore .Let is a ... more

Verified Answer ✓

From claim its enough to show that the prime field... more

Verified Answer ✓

Let be a characteristic of and characteristic of... more

Verified Answer ✓

Consider is a finite field of order , that is . ... more

Verified Answer ✓

Since is set of roots of , so if then, . This ... more

Verified Answer ✓

Since all are belongs to and linearly ... more

Verified Answer ✓

From , Let , use the binomial theorem,Each of the ... more

Verified Answer ✓

Define a map given by , use binomial theorem,Each... more

Verified Answer ✓

Prove the claim by considering the examples, first... more

Verified Answer ✓

The field is finite filed with elements, then ... more

Verified Answer ✓

From given claim its enough to show that every ... more

Verified Answer ✓

It is given that the set of roots of , , is a ... more

Verified Answer ✓

The both subfields of and of a finite field are... more

Verified Answer ✓

Let be a given field with elenments. If then ... more

Back to Top

Log In

Contact Us

Upload An Image

Please select an image to upload
Note: must be in .png, .gif or .jpg format
OR
Provide URL where image can be downloaded
Note: must be in .png, .gif or .jpg format

By clicking this button,
you agree to the terms of use

By clicking "Create Alert" I agree to the Uloop Terms of Use.

Image not available.

Add a Photo

Please select a photo to upload
Note: must be in .png, .gif or .jpg format