EXERCISE 4. Let E = F(x), the field of rational functions over F.7 (We shallhereafter denote the set of polynomials over F by F[x].) Show that the mappingƒ(x) → ƒ(x + 1) is an automorphism of the field E and prove further that thefixed elements under this automorphism are the constants (i.e., the elements of F).

Before proving the mapping f(x)→f(x+1) is an automorphism of field E, we have to know what…

Answered: EXERCISE 4. Let E = F(x), the field of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Let f(x) ∈ F[x] (F is a field) and c ∈ F. If f(x) is irreducible in F[x], then so is f(x − c).
i need help in answering this
Let R be a ring. Define a functione: R[x] ×RR by the rule e(anx" +.+ax+a0,r) = anr" + +ar+ao. In other words, given the input of a polynomial f(x) € R[x] and an element r E R, e(f(x),r) is obtained by substitutingr for x in f(x). (a) Prove that, if R is a commutative ring, then e(f,r)+e(g,r) = e(f+g.r) and e(f,r)- e(8,r) = e(f8.r) for any two linear polynomials f,g € R[x] and anyrER. [In fact both facts are true for polynomials f,g of any degree. I encourage you to write up a proof of the more general versions instead, if you can. But I didn't want to burden everyone with having to use indices and arbitrarily long sums.] (b) If R is a non-commutative ring, just one of the two equations in part (a) still holds for all r, f, and g. Which one? Explain how you could make a counterexample to the other equation. [If you know a non-commutative ring, feel free to give an actual counterexample in that ring instead of "explaining how".]
Exercise 4.2. (Bezout's identity for polynomials) Let F be a field. Let f, g E F[x] with greatest common divisor d. Prove that there exist polynomials u and v such that fu+gv = d.
Let F be a field and let f(x) be an irreducible polynomial in F[x]. Then f (x) has a multiple root in some field extension if and only if f'(x) = 0.
4) Determine whether the given set of functions is linearly independent on the interval (-x,). a) (z) = cos 2z, (2) = 1, 1,()= cos z b) (z)=z, 1,(2)=z-1, (z)=z+3 c) ()=e**, 1,(=)=e
Suppose F is a field and let p(x) € F[r]. Let f(x), g(x) € F[r] such that deg f < deg p and deg g < deg p. Show that if f(x) + (p(x)) = g(x) + (p(x)) then f(x) = g(x).
If E is a splitting field of a polynomial f(x)∈F [x] any isomorphism between subfields of E over F can be extended to an F-automorphism.
Let F be a field and a ∈ F \ {0} which is a zero of some polynomial f(x) ∈ F[x]. Find a polynomial g(x) ∈ F[x] withdeg g = deg f whose leading coefficient is the constant term of f(x) and with zero 1/a.
Please answer #10
Thank you
Let K be an extension of a field F and an element a∈K be an algebraic over F. If f(x) be a minimal polynomial of a over F, then f(x) is irreducible over F.

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS