Let E = Q(√a), where a is an integer that is not a perfect square. Show...

Let E = Q(√a), where a is an integer that is not a perfect square. Show that E/Q is normal

Expert Solution

Colby Messinger answered 2 years ago

Let A = {r belongs to Q : e < r < pi} show that A...

Let A = {r belongs to Q : e < r < pi} show that A is closed and bounded but not compact

a. Let A be a square matrix with integer entries. Prove that if lambda is a...

a. Let A be a square matrix with integer entries. Prove that if lambda is a rational eigenvalue of A then in fact lambda is an integer. b. Prove that the characteristic polynomial of the companion matrix of a monic polynomial f(t) equals f(t).

Write a Python program to find the smallest positive integer that is a perfect square and...

Write a Python program to find the smallest positive integer that is a perfect square and it contains exactly three different digits.

A positive integer N is a power if it is of the form q^k, where q,...

A positive integer N is a power if it is of the form q^k, where q, k are positive integers and k > 1. Give an efficient algorithm that takes as input a number N and determines whether it is a square, that is, whether it can be written as q^2 for some positive integer q. What is the running time of your algorithm? write the pseudocode for the algorithm.

Let E be an extension field of a finite field F, where F has q elements....

Let E be an extension field of a finite field F, where F has q elements. Let a in E be an element which is algebraic over F with degree n. Show that F(a) has q^n elements. Please provide an unique answer and motivate all steps carefully. I also prefer that the solution is provided as written notes.

(a) Let n = 2k be an even integer. Show that x = rk is an...

(a) Let n = 2k be an even integer. Show that x = rk is an element of order 2 which commutes with every element of Dn. (b) Let n = 2k be an even integer. Show that x = rk is the unique non-identity element which commutes with every element of Dn. (c) If n is an odd integer, show that the identity is the only element of Dn which commutes with every element of Dn.

6. Let t be a positive integer. Show that 1? + 2? + ⋯ + (?...

6. Let t be a positive integer. Show that 1? + 2? + ⋯ + (? − 1)? + ?? is ?(??+1). 7. Arrange the functions ?10, 10?, ? log ? , (log ?)3, ?5 + ?3 + ?2, and ?! in a list so that each function is big-O of the next function. 8. Give a big-O estimate for the function ?(?)=(?3 +?2log?)(log?+1)+(5log?+10)(?3 +1). For the function g in your estimate f(n) is O(g(n)), use a simple function g...

(a) Let <X, d> be a metric space and E ⊆ X. Show that E is...

(a) Let <X, d> be a metric space and E ⊆ X. Show that E is connected iff for all p, q ∈ E, there is a connected A ⊆ E with p, q ∈ E. b) Prove that every line segment between two points in R^k is connected, that is Ep,q = {tp + (1 − t)q | t ∈ [0, 1]} for any p not equal to q in R^k. C). Prove that every convex subset of R^k...

Let set E be defined as E={?∙?? +? | [?]∈R2}, where ?? is the natural exponential?...

Let set E be defined as E={?∙?? +? | [?]∈R2}, where ?? is the natural exponential? function, please show E is a vector space by checking all the 10 axioms. (Notice: you may use the properties of vector addition and scalar multiplication in R2)

Show that for any square-free integer n > 1, √ n is an irrational number

Question