Let f(x) and g(x) be irreducible polynomials over a field F and let a and b belong to some extension E of F. If a is a zero of f(x) and b is a zero of g(x), show that f(x) is irreducible over F(b) if and only if g(x) is irreducible over F(a).

Let f(x) and g(x) be irreducible polynomials over a field F and let a and b belong to some extension E of F. If a is a zero of f(x) and b is a zero of g(x), show that f(x) is irreducible over F(b) if and only if g(x) is irreducible over F(a).

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.3: Factorization In F [x]

Problem 8E: Let be an irreducible polynomial over a field . Prove that is irreducible over for all nonzero ...

See similar textbooks

Related questions

Q: Let F be a field and let f(x) = a,x" + a„-p"-1 + · .. Prove that x - 1 is a factor of f(x) if and…

Q: Let K be an extension of F and let a e K be algebraic over F. Then (i) There exists a unique monic…

A: Let K be an extension of F and let α∈K be algebraic over F. To prove that there exists a unique…

Q: Give an example, with justification, of a reducible separable polynomial over F125. Find the…

A: consider the equation

Q: Let F be a field and let f(r) = anr" +an-1x"-1+..+ ao € F[x]. Prove that r - 1 is a factor of f(r)…

Q: Let F be a field of characteristic 0, and let E be the splitting field of some f(x) E F[x] such that…

Q: · Let F be a field and a be a non-zero element in F. If af(x) is reducible over F, then f (x) € F[x]…

Q: -Let E be an extension field of F. Let a e E be algebraic of odd degree over F. Show that a? is…

Q: Let f(x) and g(x) be irreducible polynomials over a field F and let a and b belong to some extension…

A: Consider fx and gx be irreducible polynomials over a field F and a and b belongs to some extension E…

Q: Let K be an extension of a field F. An element a e K is algebraic over F if and only if [F (a) : F]…

Q: Prove that the only ideals of a field F are {0} and F itself.

Q: Let K be an extension of a field F. An element a e K is algebraic over F if and only if [F (a): F]…

Q: Show that if E is an algebraic extension of a field F and contains all zeros in F of every f(x) E…

A: If E is an algebraic extension of a field F and contains all zeros in F¯ of every fx∈Fx, then E is…

Q: Let & be a zero of f(x)=x2 +2x+2 in some extension field of Z3. Find the other zero of f (x) in…

A: Consider the given function fx=x2+2x+2. Note that, as the given function is quadratic, the function…

Q: Let A,B ∈Mn×n(F) be such that AB= −BA. Prove that if n is odd and F is not a field of characteristic…

A: The matrix is not invertible if its determinant is 0.The determinant of the product of the matrices…

Q: Let K be a field extension of a field F and let alpha in K. where a neo and a is algebric over F.…

Q: Let f(x) = x² +1 € Z3[x] and let R= Z3[x]/I, where I = (f(x)). (a) Show that R is a field with 9…

Q: Give an example of a polynomial ring Rx and a polynomial of degree n with more than n zeros over R.

A: A ring R is a set with two binary operations addition and multiplication that satisfies the given…

Q: Let F be a field of characteristic p > 0. Let K be the quotient field of the polynomial ring F[r].…

Q: Let F be a field. Prove that F[x]/ ≅F

Q: Prove Corollary 2 of Theorem 16.2: Let F be a field, a e F, and f (x) € F[x]. Show that a is a zero…

Q: Let F be a field. Let an irreducible polynomial f(x) ∈ F[x] be given. SHOW that f(x) is separable…

A: Let fx∈Fx be an irreducible polynomial. To prove that a polynomial f∈Fx is separable if and only if…

Q: Let F be a field. Show that there exist a, b ∈ F with the propertythat x2 + x + 1 divides x43 + ax +…

Q: 8. Let F be a field and let f(x) be an irreducible polynomial in F[x] Then if the characteristic of…

A: Let f(x) be an irreducible polynomial in Fx. The characteristic of F is zero. To show: f(x) has no…

Q: Show that if E is an algebraic extension of a field F and contains all zeros in \bar{F} of every f…

A: To show:

Q: Let f(x) belong to F[x], where F is a field. Let a be a zero of f(x) ofmultiplicity n, and write…

Q: Consider the irreducible polynomial p(x) = x² + x + 2 e Zs[x] and consider the simple extension…

Q: If F is a field and a is transcendental over F, prove that F(x) is isomorphic to F (a) as fields.

A: Please find the answer innext step

Q: Let F be a field and K a splitting field for some nonconstant polynomialover F. Show that K is a…

Q: Let f(x) be an irreducible polynomial over a field F. Prove that af(x) is irreducible over F for all…

A: Solution:Given Let f(x) be an irreducible polynomial over a field FTo prove:The function af(x) is…

Q: Let F be a field and f(x) e F[x] be a polynomial of degree > 1. If f(m =0 for some E F. thenf(x) is…

A: Since α ∈ F, x- α ∈ F[x]. Also f(x) ∈ F[x].

Q: If F is a field containing an infinite number of distinct elements, the mapping f → f~ is an…

Q: Let F be a field and let f(x) be a 5 points polynomial in F[x] that is reducible over F. Then * O…

Q: Let F be a field and let I = {a„x" + a„-1.X"-1 a, + an-1 + · ··+ ao = 0}. ...+ ao I an, an-1, . .. ,…

A: We will test following two things to check if I is an ideal of F[x] (a) For f(x),g(x) in I,…

Q: Let F be a field and let a be a non-zero element in F. If f(ax) is irreducible over F, then…

Q: Let f(x) E F[x] be irreducible. Then f(x) has a root of order greater than 1 in some extension field…

A: Given is that the f(x) ∈F(x) is reducible. We need to prove that f(x) has a root of order greater…

Q: If F = (M, N, P) and all of the components of this field are polynomials in x, y, and z, then which…

Q: Let F be a field, and let f(x) and g(x) belong to F[x]. If there is nopolynomial of positive degree…

Q: Let F be a field and let f(x) be an irreducible polynomial in F[x]. Then f (x) has a multiple root…

Q: 3. Let F be a field. Suppose that a polynomial p(x) = ao + a1x+ .+ anx" is reducible in F[x]. Prove…

A: Definition: Let (F,+,⋅) be a field and let f ∈F[x]. Then f is said to be Irreducible over F if f…

Q: Let F be a field and let a be a nonzero element of F.a. If af(x) is irreducible over F, prove that…

Q: Let f(x) EZ[x] be an irreducible polynomial of degree 4 such that it isGalois group over Q is…

Q: Show that the polynomial x® + x³ +1 is irreducible over the field of rationals Q.

A: The given polynomial is x6+x3+1.

Q: 1) Generate the elements of the field GF(2^4) using the irreducible polynomial ƒ(x) = x^4 + x^3 + 1.…

Q: Recall the following table, which shows that is a primitive element of the field GF(16)…

A: As per the question we are given a polynomial ring Z2[x] and its quotient ring Z2[x]/(x4+x+1) From…

Q: Let K be an extension of F and let a e K be algebraic over F. Then (i) There exists a unique monic…

Q: 10. An irreducible polynomial f(x) over a field of characteristic p> SECA ELM

Q: Let f (x) ∈ F[x]. If deg f (x) = 2 and a is a zero of f (x) in someextensionof F, prove that F(a) is…

Q: Let ϕ : F → R be a ring homomorphism from a field F into a ring R. Prove that if ϕ ( a ) = 0 for…

A: Consider ϕ : F → R be a ring homomorphism from a field F into a ring R,Since, ϕ ( a ) = 0 for some…

Q: Attached is the question I'm needing help with answering. TIA!

A: Note: For simplicity, we have used α as a zero rather than a as a zero. Which writing the answer we…

Q: Let F be a field. Given an irreducible polynomial f(x) ∈ F[x] with f'(x) (x) not equal 0, SHOW that…

Question

Please include reasons for each step. Thanks!

34. Let f(x) and g(x) be irreducible polynomials over a field F and let
a and b belong to some extension E of F. If a is a zero of f(x) and
b is a zero of g(x), show that f(x) is irreducible over F(b) if and only
if g(x) is irreducible over F(a).

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Similar questions

Prove Theorem If and are relatively prime polynomials over the field and if in , then in .
Suppose that f(x),g(x), and h(x) are polynomials over the field F, each of which has positive degree, and that f(x)=g(x)h(x). Prove that the zeros of f(x) in F consist of the zeros of g(x) in F together with the zeros of h(x) in F.
Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.
Let be an irreducible polynomial over a field . Prove that is irreducible over for all nonzero in
Let F be a field and f(x)=a0+a1x+...+anxnF[x]. Prove that x1 is a factor of f(x) if and only if a0+a1+...+an=0. Prove that x+1 is a factor of f(x) if and only if a0+a1+...+(1)nan=0.
Let be a field. Prove that if is a zero of then is a zero of
Prove Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros in
Let where is a field and let . Prove that if is irreducible over , then is irreducible over .
Use Theorem to show that each of the following polynomials is irreducible over the field of rational numbers. Theorem Irreducibility of in Suppose is a polynomial of positive degree with integral coefficients and is a prime integer that does not divide. Let Where for If is irreducible in then is irreducible in .
If is a finite field with elements, and is a polynomial of positive degree over , find a formula for the number of elements in the ring .
Prove Theorem Suppose is an irreducible polynomial over the field such that divides a product in , then divides some .
True or False Label each of the following statements as either true or false. 8. Any polynomial of positive degree that is reducible over a field has at least one zero in .