Please see the attached file and explain it step by step so I can understand the work. Thanks!
Let F be a field, and let I be an ideal of F. Let O be the zero ring. Prove that F/I isomorphic O or F/I isomorphic F.
Check if the proof is correct. I need help to justify some of my answers by using Theorems, Definitions and etc. You can change my wording but try to stick to my idea. It's really important that you explain your work. Thanks! Note: R=Real Number and C=Complex Number
A student thinks of a polynomial p(x) of arbitrary degree, and non-negative integer coefficients.
A student thinks of a polynomial p(x) of arbitrary degree, and non-negative integer coefficients. How can you determine the student's polynomial by asking for two values of her polynomial, say p(a) and p(b), where a and b are positive integers? Hint: A positive integer n can be written uniquely in base k, where k is a positive ...continues
Find to three decimal places the one real root of X^3 + 3X^2 + 2 = 0. Then use the approximate real root and compute the two conjugate roots using the graphical method of Yanosik.
Let p1(x) = x^3 + x + 1 and p2(x) = x^3 + s + 2 in F5[x].
I really need some help with this. Someone has got to have the background for this one. "Let p1(x) = x^3 + x + 1 and p2(x) = x^3 + s + 2 in F5[x]. F5[x] is ust the set of all polynomials in x with coefficients from the set {0,1,2,3,4} with arithmetic done mod 5. Compute (x+2)^2114 in F5[x]/(p1(x)) and in F5[x]/(p2(x)). The ...continues
Archimedes found his approximation to Pi by inscribed polygons
Archimedes found his approximation to Pi by inscribed polygons - Start with a circle of radius 1 and a suitable starting polygon and solve for an inscribed polygon of 96 sides. Next, start with a unit circle and appropriate inscribed regular polygon to approximate Pi using a regular inscribed n-gon with 4096 sides.
Proofs (see attached)
1. Prove that the polynomial x^4 - 16*(x^2) + 4 is irreducible in Q[x] (the ring of all polynomials with rational coefficients). 2. Show that every element of C (i.e., every complex number) is algebraic over R (the ring of all real numbers).
Abstract Algebra: Algebraic Extensions
Please see the attachment to see these questions properly. _____________________________________________ Question 1 If [K:F] is finite and u is algebraic over K ,prove that [F(u):F] divides [K(u):F] Hint:[F(u):F] and [K(u):F(u)] are finite by Theorems 10.4,10.7 and 10.9 Apply Theorem 10.4 to Theorem 10.4 Le ...continues
