Answered: Theorem 22.3 Subfields of a Finite…

Theorem 22.3 Subfields of a Finite Field For each divisor m of n, GF(p") has a unique subfield of order p". Moreover, these are the only subfields of GF(p").

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.3: The Characteristic Of A Ring

Problem 18E

See similar textbooks

Related questions

Q: Find the smallest field that has exactly 6 subfields.

Q: Find the irreducible tactors of 2++² + 2x +1 in Zs[z] For which of the primes p E {3,5} can we…

A: (a). Given polynomial is fx=x5+x4+x2+2x+1 in ℤ3x. Therefore the given polynomial can be written as:…

Q: Give an example of a polynomial of degree 4 over the field R of real numbers that is reducible over…

A: Solution: Consider we have a polynomial having degree 4 then it must be reducible over R (field of…

Q: Show that x^2 + x + 1 is irreducible over Z_2 and has a zero in some extension field of Z_2 that is…

A: First, to show the polynomial x2 +x+1 is irreducible over Z2:Here, recall that a polynomial of…

Q: . Show that x2 + 3 and x2 + x + 1 over Q have same splitting field.

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: 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: What are the possible orders of the elements of the finite field F25 of 25 elements ?

Q: Find the splitting field of x3 - 1 over Q. Express your answer in theform Q(a).

Q: 1. What are the possible automorphism groups of the splitting field of an irreducible polynomial of…

A: Introduction: Galois group is one of the very significant group in extended field theory. In every…

Q: 2. (a) Why is the polynomial X° - 2 irreducible over Q ? What is its splitting field K and what is…

A: Content of a polynomial- Let R be a unique factorization domain (UFD) and let f(x) be a non-zero…

Q: It is possible to write z² + 6z + 4 in Zu[z] with the help of linear factors. Find this…

A: (a). The given polynomial is px=x2+6x+4 in ℤ11x. Observe that p1=12+61+4=11≈0 Hence the polynomial…

Q: Q1: Prove that every finite integral domain is field?

Q: Find the smallest field of characteristic 2 that contains an elementwhose multiplicative order is 5…

Q: Suppose that E is the splitting field of some polynomial over a field Fof characteristic 0. If…

Q: Establish the following assertion there by completing the proof of Theorem 3-28: If (F , +,.) is a…

A: (F,+,.) is a field of characteristic zero and (K,+,.) the prime subfield generated by the identity…

Q: Show that Z3[x]/<x2 +x + 1> is not a field.

Q: Prove that the characteristic of a field is either or a prime.

A: We need to prove : The characteristic of a field is either 0 or a prime W.k.t if the field has…

Q: Show that no finite field is algebraically closed.

Q: Let F be a field of order 32. Show that the only subfields of F areF itself and {0, 1}.

Q: Find all polynomials p(x) E Z2[x] of degree at most 3 such that Z2[x]/{p(x)) is a field. How many…

Q: 1) Z12/1 is not a Field Always because if we take the ideal I = Z₁2/1 is a Field. 2) The map y: Z₁-…

A: Given , f(x)=x6+5x5-15x4+15x3+25x2+5x+25 (.) Eisenstein' s Criterion for…

Q: Can a Irreducible polynomial be a field with a degree that is larger or the same as 1?

A: Can a Irreducible polynomial be a field with a degree that is larger or the same as 1?

Q: Can a field with 128 elements contain a subfield with 8 elements? Give an explicit construction of…

Q: Abstract Algebra

A: To define the concept of a subfield of a field and prove the stated property regarding subfields of…

Q: Suppose that E is the splitting field of some polynomial over a fieldF of characteristic 0. If…

Q: Suppose f(x) E Z,[r] and f(x) is irreducible over Z,, where p is prime. (a) If deg f(x) = n, prove…

Q: For which n listed below does there exist a field extension F ɔ Z/2 of degree n such that the…

A: Follow the procedure given below.

Q: Show that the operation of multiplication defined in the proof ofTheorem 15.6 is well-defined

Q: Let m and n be relatively prime positive integers. Prove that thesplitting field of xmn - 1 over Q…

Q: Prove that a field with 16 elements contains a subfield with 4 elements.

Q: not a Field Always because if we take the ideal I = : Z₂ —————→ Z₂ such that y(x) = { 1 if x is odd…

Q: 3. (a) Define i. A finite field extension. ii. An algebraic element of a field extension. iii.…

A: 3. (a) To Define: i. A Finite Field extension. ii. An algebraic element of a field extension. iii.…

Q: Prove that in a field, the multiplicative identity 1 is always unique.

A: Consider a field F under the standard operation addition and multiplication. An element e of F is…

Q: Prove that no order can be defined in the complex field that turns it into an ordered field. (Hint:…

Q: 2. Suppose F is a finite field. Prove that F× is cyclic. Deduce that x² = -1 has a solution in a…

A: Here, F× denotes F-0. We need to prove F× is a cyclic group under multiplication.

Q: Show that a finite extension of a finite field is a simple extension

Q: n. 1, a. Show that there exists an irreducible polynomial of degree 3 in Z3[x]. b. Show from part…

A: a) Let If , then If , then If , then So, has no zero in

Q: C. Prove that F2 (as defined on p20 of the notes) is a field.

Q: If F is a Field it is known that a polynomial f(x) of degree 2 or 3 in F[x] is irreducible if and…

A: We have to show that a polynomial f(x) of degree 4 has no zeros in a field F but f(x) is not…

Q: Show that the cubic field generated by a root of f(x) = %3D x3 -3x2 - 10x - 8, where f is…

A: To show that the cubic field generated by roots of fx=x3−3x2−10x−8 where f is irreducibe over Q, and…

Q: Show that R[x]/ is a field.

A: In this question, we have to show that ℝxx2+1 is a field.

Q: 7. Show that the splitting field Q(√2, √3) of (X2-2)(X2-3) is a Galois extension of the rational…

A: If there exist a polynomial of degree 4 generated by this splitting field, then we say this is an…

Q: 7- If f E F[xis irreducible polynomial, then the field E can be viewed as a subfield of a field…

Q: A field of characteristic p *0 such that each element of the field is the ph power of some member of…

Q: F is

A: Given: A field F.We have to show that a monic polynomial in F[x] can be factored as a product of…

Q: Abstract Algebra

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

Q: (3) (a) Suppose a ring R is a finitely generated algebra over a field k. Prove that the Jacobson…

Q: Find all monic irreducible polynomials of degree 2 over Z3.

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

Question

Prove the uniqueness portion of Theorem 22.3 using a group theoretic argument.

Theorem 22.3 Subfields of a Finite Field
For each divisor m of n, GF(p") has a unique subfield of order p".
Moreover, these are the only subfields of GF(p").

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

Step 3

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Groups$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra 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
Prove that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.
Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.
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 a0 in a field F, prove that for every bF the equation ax=b has a unique solution x in F. [Type here][Type here]
Prove Theorem If and are relatively prime polynomials over the field and if in , then in .
Since this section presents a method for constructing a field of quotients for an arbitrary integral domain D, we might ask what happens if D is already a field. As an example, consider the situation when D=5. a. With D=5, write out all the elements of S, sort these elements according to the relation , and then list all the distinct elements of Q. b. Exhibit an isomorphism from D to Q.
Prove that if R and S are fields, then the direct sum RS is not a field. [Type here][Type here]
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]
Corollary requires that be a field. Show that each of the following polynomials of positive degree has more than zeros over where is not a field. over over
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 .)
Prove that every ordered integral domain has characteristic zero.