Exercise 3 Let E, F be subfields of C and suppose E = F(a) for some a E E. Write f(x) E Flxl for ...

Question

Question

Exercise 3 Let E, F be subfields of C and suppose E = F(a) for some a E E. Write f(x) E Flxl for the minimal polynomial of a

Add a comment Improve this question Transcribed image text

Answer 1

Answer #1

ir fcn] factors into linead fa co«s in Tha s E ontains al he Tools of om), we wan eE a nor mal enension of F LEF de eej wan t Gndl Since Ea enten sion o F Field or The Smallest Fre lol G. nlaining g and FCd)らECA) Oven FCd anal i lat Gnol maps d and EC

Add a comment

Answer 2

Similar Homework Help Questions

Let E = F(a) be a (simple) extension of F. wherea E E is algebraic over F. Suppose the degree of ...

Let E = F(a) be a (simple) extension of F. wherea E E is algebraic over F. Suppose the degree of α over F is n Then every β E E can be expressed uniquely in the form β-bo-b10 + +b-1a-1 for some bi F. (a) Show every element can be written as f (a) for some polynomial f(x) E F (b) Let m(x) be the minimal polynomial of α over F. Write m(x) r" +an-11n-1+--+ n_1α α0. Use this...
Please show all steps. Linear Algebra Bonus: Let A E Mn(C) such that A0 for some...

Please show all steps. Linear Algebra Bonus: Let A E Mn(C) such that A0 for some positive integer m. Prove that An 0 Hint: What is the minimal polynomial for A?
Consider the polynomial f(x) = x p − x + 1 ∈ Zp[x]. (a) Let a be a root of f in some extension. Prove that a /∈ Zp and a + b is a root of f for all b ∈ Zp. (b) Prove that f is irreducible over Zp. [Hi...

Consider the polynomial f(x) = x p − x + 1 ∈ Zp[x]. (a) Let a be a root of f in some extension. Prove that a /∈ Zp and a + b is a root of f for all b ∈ Zp. (b) Prove that f is irreducible over Zp. [Hint: Assume it is reducible. If one of the factors has degree m, look at the coefficient of x m−1 and get a contradiction.]

1. Let ab and f E C[a, b], and let E(0, ))- - (co +c)w(a) da for some weight function w(x) >0. (a) Use calculus to w...

1. Let ab and f E C[a, b], and let E(0, ))- - (co +c)w(a) da for some weight function w(x) >0. (a) Use calculus to write down a linear system for the critical point of E(co, c1). (b) Is the solution of this linear system the same as that of the normal equations arising from the use of Theorem 2 on page 395 to optimize co, ci under the norm 1/2 ? (c) Use your results to find the...
Exercise 1 Let E, FCC and let E 2F be a field. (a) Without using the primitive element theorem, s...

Exercise 1 Let E, FCC and let E 2F be a field. (a) Without using the primitive element theorem, show that [E : FI < oo if and only if (Hint: Tower law → [F(01, , , , a.) : F1 < oo. For the other direction, use induction on IE、F) (b) Suppose that E F(a Show that E is normal if and only if for every E Flai, . . . , α.) for some on. . . ....
6. (16 points) Let CE C be a primitive n-th root of unity. Let X =...

6. (16 points) Let CE C be a primitive n-th root of unity. Let X = 6 + 1/5. (a) (4 points) Show that Q(5) R = Q(1). (b) (4 points) Let f be the minimal polynomial of over Q. Show that Q(x) is a splitting field of f over Q. (c) (4 points) Show that Gal(Q(^)/Q) – (Z/nZ)* / (-1). (d) (4 points) Find the minimal polynomial of 2 cos(27/9) over Q.

8. Let n be a positive integer. The n-th cyclotomic polynomial Ф,a(z) E Z[2] is defined recursively in the following way: 1. Ф1(x)-x-1. 2. If n > 1, then Фп(x)- , (where in the product in the...

8. Let n be a positive integer. The n-th cyclotomic polynomial Ф,a(z) E Z[2] is defined recursively in the following way: 1. Ф1(x)-x-1. 2. If n > 1, then Фп(x)- , (where in the product in the denomina- tor, d runs through all divisors of n less than n). . A. Calculate Ф2(x), Ф4(x) and Ф8(z): . B. n(x) is the minimal polynomial for the primitive n-th root of unity over Q. Let f(x) = "8-1 E Q[a] and ω...
1. Let T be the matrix T=10 3 acting on the complex vector space V C3 (a) Recall how T defines the structure of a C-mod...

1. Let T be the matrix T=10 3 acting on the complex vector space V C3 (a) Recall how T defines the structure of a C-module on C3. (b) Let p(x71, and let 2Compute the element p(x) v of C3 (c) Give a set of generators and relations for C3 over Cz] with the above module structure. (d) Write down the relations matrix (e) Recall the definition of minimal polynomial of a matrix. (f) What is the minimal polynomial of...
Let k C F = kla) be a simple algebraic extension. Prove that F is normal...

Let k C F = kla) be a simple algebraic extension. Prove that F is normal over k if and only if for every algebraic extension FC K and every o E Autk(K),(F) = F.

Exercise 25: Let f: [0,1R be defined by x=0 fx)/n, m/n, with m, n E N and n is the minimal n such that z m/n x- m/n, with m,n E N and n is the minimal n such that x a) Show that L(f, P) = 0 for all p...

Exercise 25: Let f: [0,1R be defined by x=0 fx)/n, m/n, with m, n E N and n is the minimal n such that z m/n x- m/n, with m,n E N and n is the minimal n such that x a) Show that L(f, P) = 0 for all partitions P of [0, 1]. b) Let m E N. Show that the cardinality of the set A :-{х є [0, 1] : f(x) > 1/m} is bounded by m(m...

Exercise 3 Let E, F be subfields of C and suppose E = F(a) for some a E E. Write f(x) E Flxl for ...

Homework Answers

Add Answer to:
Exercise 3 Let E, F be subfields of C and suppose E = F(a) for some a E E. Write f(x) E Flxl for ...

Post as a guest

Earn Coins

Let E = F(a) be a (simple) extension of F. wherea E E is algebraic over F. Suppose the degree of ...

Please show all steps. Linear Algebra Bonus: Let A E Mn(C) such that A0 for some...

Consider the polynomial f(x) = x p − x + 1 ∈ Zp[x]. (a) Let a be a root of f in some extension. Prove that a /∈ Zp and a + b is a root of f for all b ∈ Zp. (b) Prove that f is irreducible over Zp. [Hi...

1. Let ab and f E C[a, b], and let E(0, ))- - (co +c)w(a) da for some weight function w(x) >0. (a) Use calculus to w...

Exercise 1 Let E, FCC and let E 2F be a field. (a) Without using the primitive element theorem, s...

6. (16 points) Let CE C be a primitive n-th root of unity. Let X =...

8. Let n be a positive integer. The n-th cyclotomic polynomial Ф,a(z) E Z[2] is defined recursively in the following way: 1. Ф1(x)-x-1. 2. If n > 1, then Фп(x)- , (where in the product in the...

1. Let T be the matrix T=10 3 acting on the complex vector space V C3 (a) Recall how T defines the structure of a C-mod...

Let k C F = kla) be a simple algebraic extension. Prove that F is normal...

Exercise 25: Let f: [0,1R be defined by x=0 fx)/n, m/n, with m, n E N and n is the minimal n such that z m/n x- m/n, with m,n E N and n is the minimal n such that x a) Show that L(f, P) = 0 for all p...

Exercise 3 Let E, F be subfields of C and suppose E = F(a) for some a E E. Write f(x) E Flxl for ...

Homework Answers

Add Answer to: Exercise 3 Let E, F be subfields of C and suppose E = F(a) for some a E E. Write f(x) E Flxl for ...

Post as a guest

Earn Coins

Add Answer to:
Exercise 3 Let E, F be subfields of C and suppose E = F(a) for some a E E. Write f(x) E Flxl for ...