Answered: Let F be a field and let a be a…

Let F be a field and let a be a non-zero element in F. If f(ax) is irreducible over F, then f(x)EF[x] is * Unit Reducible None of the choices Irreducible

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.4: Zeros Of A Polynomial

Problem 33E: Let where is a field and let . Prove that if is irreducible over , then is irreducible over .

See similar textbooks

Related questions

Q: Let K be an extension of a field F. If an) is a finite an e K are algebraic over F, then F (a1, a2,…

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: 30. Let E be an extension field of a finite field F, where F has q elements. Let a e E be algebraic…

A: The objective is to prove that, if is a finite field and has elements. If be an extension field…

Q: Abstract Algebra. Please explain everything in detail.

A: To prove the stated properties pertaining to the subfields of the given field

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 4 elements and let f(X) e F[X] be an irreducible polynomial of degree 4. How…

A: Given that F is a field of 4 elements. f(x) ∈ F[x] is an irreducible polynomial. then we have to…

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: Suppose F is a field of characteristic p. Show that F can be regarded as a vector space over GF(p).…

A: Given F is a field of characteristic p. For k∈GFp, x∈F. For F to be a vector space the product is…

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 E be an extension of field of F. Let α ∈ E be algebraic of odd degree over F. Show that α2 is…

A: let E be an extension of field of F.Let α∈E be algebraic of odddegree over F.Show that α2 is an…

Q: Let x, y ∈ F, where F is an ordered field. Suppose 0 < x < y. Show that x2 < y2.

A: multiplying by x and y in x<y respectively and then comparing both result.....

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 F be a field and aeF be such that [F (a): F]=5. Show that F(a)= F(x³).

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

A: Use the properties of ring of polynomials to solve this problem.

Q: Prove that every field is an integral domain, but the converse is not always true. (Hint: See if…

A: Let F be a any field. Therefore, F is commutative ring with unity and possess multiplicative inverse…

Q: Let F be a finite field with n elements. Prove that xn-1 = 1 for allnonzero x in F.

Q: et f(x) in Fla] be a nonconstant polynomial and let K and L be its splitting field over F. Then…

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

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

Q: 31 Let F be a field and let f(x) in F[x] be a nonconstant polynomial. Let K be the splitting field…

A: Given F be a field and let f(x) in F[x] be a nonconstant polynomial. Let K be the splitting field of…

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

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

Q: et F be a field and let a be a bé ero element in F. If f(ax) is reducible over F, then f(x)

A: In the given question, the concept of the irreducible polynomial is applied. Irreducible Polynomial…

Q: Let (S, +,) be a subfield of the field (F, +,), then (S, +,) is a) integral domain b) field c)…

A: Hello, learner we can answer first question as per the honor policy. Please resubmit other question…

Q: Let (F, +, ·, <) be an ordered field. Show that for any y ∈ F, the equation x^3 = y has at most…

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 F, 2/2Z. Find an irreducible polynomial of degree 4 in Fla] and use it to construct a field…

A: In this question, it is given that F = F2 = ℤ2ℤ . We have to find the irreducible polynomial of…

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: Let f be a polynomial over the field F with derivative f'. Then f is a product of distinct…

A: It is given that f is a polynomial over the field F with derivative f'. The objective is to show…

Q: Suppose that F is a field and every irreducible polynomial in F[x] islinear. Show that F is…

Q: Consider the integral domain D = {x+yv2: x, y ≤ Z}. (a) Apply the construction of field of quotients…

A: The given question is related with abstract algebra. Given the integral domain D = x + y2 : x , y ∈…

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

Q: Let F be a field. Prove that Fl) E F.

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 A C B be a finite extension of fields whose degree is 17. Show that there is no intermediate…

A: Let A⊆B be a finite extension of fields whose degree is 17 which is prime . An extension B over A is…

Q: if a field F has order n, then F* has order n-1

A: There is a theorem that says , "If a field F has order n, then F* has order n-1". Statement of…

Q: Let S = {0, 1} and F = R, the field of real numbers. In F(S, R), show that f =g and f +g=h, where…

A: Given:S = {0, 1} and F = R, the field of real numbers.To show that f =g and f +g=h, where f(x) = 2x…

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

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

Q: (B) Explain the relationship between each of th. a) Field and integral domain. b)

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: 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: Show that Let K be an extension of a field F and a e K be algebraic over F. Then F[a] = F (a), where…

Q: Find the field of quotients of integeral domain z[i] and z[√2]

A: We will solve this question using basics of ring theory and just the definition of Fields and…

Q: Abstract Algebra

A: To prove the existence of infinitely many monic irreducible polynomials over any given field F.

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

Q: Let F be a field and let f(x) be a polynomial in F[x] that is reducible over F. Then * O is not a…

A: Let F be a field. We say that a non-constant polynomial f(x) is reducible over F or a reducible…

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: 8. Let f: R-→R be a field homomorphism. Show that f is identity.

A: Introduction: Like integral domain, a field also have homomorphism. A map f:F→K is referred to as…

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

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.

Question

<x-2>
<x-1>
Let F be a field and let a be a non-zero element in F. If f(ax) is irreducible
over F, then f(x)EF[x] is *
Unit
Reducible
None of the choices
Irreducible
+x³ + x² + x + 1 is irreducible over
88 F
to search
近

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 with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Application of Differentiation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Prove Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros in
Suppose S is a subset of an field F that contains at least two elements and satisfies both of the following conditions: xS and yS imply xyS, and xS and y0S imply xy1S. Prove that S is a field. This S is called a subfield of F. [Type here][Type here]
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.
8. Prove that the characteristic of a field is either 0 or a prime.
Let be an irreducible polynomial over a field . Prove that is irreducible over for all nonzero in
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 .
Prove that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.
Let where is a field and let . Prove that if is irreducible over , then is irreducible over .
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.
Each of the polynomials in Exercises is irreducible over the given field . Find all zeros of in the field obtained by adjoining a zero of to . (In Exercises and , has three zeros in .)
Let be a field. Prove that if is a zero of then is a zero of
Let S be a nonempty subset of an order field F. Write definitions for lower bound of S and greatest lower bound of S. Prove that if F is a complete ordered field and the nonempty subset S has a lower bound in F, then S has a greatest lower bound in F.