Let K be a finite normal extension of F. a. For a K, show that is

Question:

Let K be a finite normal extension of F.

a. For a ∈ K, show that

is in F[x]

b. Referring to part (a), show that f(x) is a power of irr(α, F), and f(x) = irr(α, F) if and only if K = F(α).

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For book-img-for-question

A First Course In Abstract Algebra

ISBN: 9780201763904

7th Edition

Authors: John Fraleigh

See More Books

Question Posted: Jan 16, 2023 02:50 AM

See More Questions

Let K be a finite normal extension of a field F. Prove that for every K, the norm of over F, given by and the trace of over F, given by are elements of F. Nx/F(a) = (), G(K/F)
Let K be a vector space over Z3 with basis {1, t}, so K = {a + bt | a, b in Z3}. It is known that K becomes a field of nine elements if we define t2 = -1 in Z3. In each case find the inverse of the...
Let K be a positive definite 2 2 matrix. (a) Explain why the quadratic equation xT Kx = 1 defines an ellipse. Prove that its principal axes are the eigenvectors of K, and the semi-axes are the...
Why does allocating an array of length $n$ take time proportional to $n$ ?
Security Specialists performs security services for local businesses. During 2011, Security Specialists performed $915,700 of security services and collected $930,000 cash from customers. Security...
Ralph Henwood was paid a salary of $64,600 during 20-- by Odesto Company. In addition, during the year Henwood started his own business as a public accountant and reported a net business income of...
Victor and Vincents partnership had a final profit for the year of $$40$ 500. When the partnership was formed at the beginning of the year Victor invested $$150$ 000 and Vincent invested $$100$...
Monohan Machinery specializes in developing weedharvesting equipment that is used to clear small lakes of weeds. George Monohan, president of Monohan Machinery, is convinced that harvesting weeds is...
calculate both intrinsic value and time value for the options that you have picked. ANZ GROUP HOLDINGS LIMITED ANZ LAST PRICE / TODAY'S CHANGE $25.280+$0.350 (1.403 %) Industry Group: Banks VOLUME...
Let F be a field and let be a primitive nth root of unity in F, where the characteristic of F is either 0 or does not divide n. a. Show that F() is a normal extension of F. b. Show that G(F(/F) is...
Find the fixed field of the automorphism or set of automorphisms of E. { 2 , 3, 5 }
What do global organizations like the World Trade Organization and the International Monetary Fund do?
Calculate the amount of the estimated accounts receivable that the Company would report in its budgeted balance sheet at November 30, 2023. Accounts Receivable @ November 30, 2023 (4 points)
Spalding Company has offered to sell to Ping Company its assets at their book values plus $1,800,000 representing payment for goodwill. Operating data for 2023 for the two companies are as follows:...
South Dakota v. Wayfair, Inc., 138 S. Ct. 2080 (2018)\ FACT SUMMARY South Dakota, like many states, taxes the retail sales of goods and services in the state. Sellers are required to collect and...
If A is the matrix associated to the system (S), so that the system can be written in the form -( Ak X Y Z -k h+k 2h + 1 the linear transformation lk: R IR associated to Ag with respect to the...
7) A blob of clay of mass M is propelled upward from a spring that is initially compressed by an amount d. The spring constant is k. What is the ultimate height h above the unstretched spring's end...
A differential leveling loop started and closed on BM Juno, elevation 5007.86 ft. The plus sight and minus sight distances were kept approximately equal. Readings (in feet) listed in the order taken...
Why do CPA firms sometimes use a combination of positive and negative confirmations on the same audit?
You should have seen that 0.50 is a plausible value for the true proportion of all 16- to 17-year-olds who have used their cell phones while driving. Does this prove that 0.50 is the true value? Lets...
Use the One Proportion applet to create an interval of plausible values with 95% confidence for the population proportion of drivers 16 to 17 who have ever talked on a cell phone while driving. Do...
Now use the One Proportion applet to create an interval of plausible values with 95% confidence for the population proportion of drivers 16 to 17 who have ever talked on a cell phone while driving...
Given that real numbers a,b,c are all not zero, and a+b+c=0. Find the value of x^(2007)-2007x+2007, where x=(|a|)/(b+c)+(|b|)/(a+c)-(|c|)/(a+b).
Which cash flow is riskier? the residual/disposition value the annual lease payments?
Please answer! 3 points A T-bill that is 210 days from maturity is selling for $97,000. The T-bill has a face value of $100,000. 1. Calculate the discount yield. 2. Calculate the bond equivalent...

Let K be a finite normal extension of F. a. For a K, show that is

Question:

f(x) = Π (-σ(α)) σεG(K/F)

Step by Step Answer:

a For GKF we have a natural extension of to an automorphism of Kx where 0 1 x n x ...View the full answer

A First Course In Abstract Algebra

Students also viewed these Mathematics questions