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

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. 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).

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 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.

(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...

(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...

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)

Let D8 be the group of symmetries of the square. (a) Show that D8 can be...

Let D8 be the group of symmetries of the square. (a) Show that D8 can be generated by the rotation through 90◦ and any one of the four reflections. (b) Show that D8 can be generated by two reflections. (c) Is it true that any choice of a pair of (distinct) reflections is a generating set of D8? Note: What is mainly required here is patience. The first important step is to set up your notation in a clear way,...

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