Extension Definition and 279 Threads

Hi, I've been asked to find $[ \mathbb{Q}(\sqrt{2}, i ): \mathbb{Q} ]$, and to write down a basis. Now I know that $[ \mathbb{Q}(\sqrt{2}, i ): \mathbb{Q} ] = 4$, and that a basis could be $ \left \{ 1, i, \sqrt{2}, i\sqrt{2}\right \} $ it is whether the way I am explaining how I arrived here...

Homework Statement Let L_1/K,\,L_2/K be extensions of p-adic fields, at least one of which is Galois, with ramification indices e_1,\,e_2 . Suppose that (e_1,\,e_2) = 1 . Show that L_1 L_2/K has ramification index e_1 e_2. Homework Equations I have most of the proof done: I'm trying to show...

Homework Statement Let F|K be a field extension. If v e F is algebraic over K(u) for some u e F and v is transcendental over K, then u is algebraic over K(v).Homework Equations v transcendental over K implies K(v) iso to K(x). Know also that there exists f e K(u)[x] with f(v) = 0.The Attempt at...

Hungerford says that But if we take K = ℝ and K(x_{1}) = ℝ(i) = ℂ, we have that i is not in ℝ yet is algebraic over ℝ. Guess I'm missing something here. Is it that this need not be true for simple extensions if the primitive element is algebraic over the field?

how to extend a locally defined function to a smooth function on the whole manifold ,by using a bump function?

Homework Statement 2 blocks, mass 'M1' & 'M2' are connected by an ideal spring of force constant 'k' and placed on a frictionless surface. Force 'F' is applied on the 'M2' block. We have to find the maximum extension in the spring. 2. The attempt at a solution (1) The conservation of energy...

I am referring to this: A brainwave hit me and I wondered the following: If the moment when i unplug or switch it off (In my country they have a switch for added safety reason), the live wire happens to be at the peak of the AC curve, will my extension cord effectively become similar to a...

1. The problem statement 1)Do electric fields extend through a vacuum? 2)Do electric fields extend through the interior of a insulator? 3)Do electric fields extend through the interior of a conductor? I am just starting to understand electric fields but I am still very unsure, and am not...

Homework Statement Bungee Jumping l=natural length of rope x=extension of rope y=total distance fallen m=mass,a=acceleration,v=velocity,g=acceleration due to gravity k=air resistance co-efficient Given data: The rope is stretched to twice its natural length by a mass of 75kg hanging at rest...

A spring of mass M is suspended from the ceiling of a room. Find the extension in the spring due to its own weight if it has a spring constant of value k . I am getting answer as \frac{Mg}{2k} , but the answer given in back of the book is \frac{Mg}{3k}. What I did was : Let the natural...

My group did an experiment on rubber bands. The rubber bands were stretched multiple times from 50, 100 and 150 times and then a weight was attached to the rubber band to record its extension. On average the rubber band's extension increased by 10 cm after each increment of 50 stretches. I...

By F[X] I mean the polynomials with coefficients in field F. By F(X) I mean the rational polynomials. I have a feeling that \boxed{ \mathbb Q( \sqrt 2 ) \cong \frac{\mathbb Q[X]}{(X^2-2)}} . [SIZE="2"](if not readable: the RHS is with [X]) Is this true? If so, how can I prove it? I suppose...

Hello, Say we have field (F,+,.) and field extension (E,+,.), then the degree of the field extension (i.e. the dimension of the vector field E across the field F) is given the symbol [E:F] . But we can also see F and E as the groups (F,+) and (E,+), and then the same symbol denotes the...

Homework Statement Show that p(x) = x^2 - \sqrt2 is irreducible in \mathbb Z[\sqrt 2] . The Attempt at a Solution I think I have this, but I just want to make sure my reasoning is correct. I'm sure there are other ways. Firstly, it is sufficient to show that p(x) is irreducible in the...

I do not understand the proof of Proposition 0.16 in Allen Hatcher's book Algebraic Topology. If someone has the book, could you please clarify the part of the proof when he says "If we perform the deformation retraction of X^n\times I onto X^n\times\{0\}\cup (X^{n-1}\cup A^n)\times I during...

Homework Statement Let E be an extension of a finite field F, where F has q elements. Let \alpha \epsilon E be algebraic over F of degree n. Prove F \left( \alpha \right) has q^{n} elements. Homework Equations An element \alpha of an extension field E of a field F is algebraic over F if f...

Ok, the book I'm reading states Gauss's lemma as such: If f(x) is a monic polynomial with integral coefficients that factors into two monic polynomials with coefficients that are rational, f(x) = g(x)h(x), then g(x), h(x) \in \mathbb{Z}[x]. Now one of the exercises says to prove that: If...

1.Let F be an extension field of K and let u be in F. Show that K(a^2)contained in K(a) and [K(u):K(a^2)]=1 or 2. 2.Let F be an extension field of K and let a be in F be algebraic over K with minimal polynomial m(x). Show that if degm(x) is odd then K(u)=K(a^2). 1. I was thinking of...

c++ header files with and without "h" extension hi fine people,:wink: i have seen some people use header files with ".h" extension and some without it. what is reason? i think there are some header files where you are required to use ".h" such as "#include <windows.h>. you can not use...

1.Let F=K(u) where u is transcedental over the field K. If E is a field such that K contained in E contained in F, then Show that u is algebraic over E. Let a be any element of E that is not in K. Then a = f(u)/g(u) for some polynomials f(x), g(x) inK[x] 2.Let K contained in E...

I've been seeing this certain file extension lately on my computer, but I do not know what programs to run them. What are file extension dll?

Homework Statement The Attempt at a Solution first using the initial extension i found out that k = 200. velocity of block = √(2gh) = √(2*10*.45) = 3 then using conservation of momentum ... 120 * 3 = 320 * V V = 1.125 this kinetic energy converts into potential energy of...

Does anyone know of an equation that models the build up of potential energy in an extension spring induced by twisting it when its motion is otherwise constrained in the transverse direction?

I'm surprised no one has posted a thread on this as yet. Talkita is a Google Chrome extension that allows you to talk to people who are on the same website as you. Check it out! https://chrome.google.com/webstore/detail/gffkbdneeepeccpbpoaciomdibieaiii This way, I can chat with whoever is...

Homework Statement Show that the elements 3 and 2+\sqrt{-5} in \textbf{Z}[\sqrt{-5}] have a greatest common divisor of 1, but the ideal I = (3, 2+\sqrt{-5}) is not the total. Conclude that I is not principle. Homework Equations The Attempt at a Solution I have got as far as...

The equation of motion of a rocket with mass depletion during ascent and subject to drag forces can be written as M(t) dV/dt = A - M(t)g - BV^2 (Eq. 1) with initial condition V(t=0) = 0 (V is velocity and t is time) Let us assume a linear mass depletion according to...

Homework Statement Let K be the subfield of all constructible numbers in C Let A be the subfield of all algebraic numbers in C Is the field extension A:K finite? The Attempt at a Solution I don't know where to start! I have read and understood proofs that all constructible numbers...

[b]1. Collagen fiber is stressed with 12 N force. The cross-sectional area of the fiber is 3 mm2, its coefficient of elasticity is 500 MPa. Give the percentage of relative extension. I think i have found the correct equation but i can't seem to find the rigth answer which is suposed to be...

evaluate the value of extension ( x) which elongate under the action of load (w) of 6 N ? [Use your results obtained from rest 1 ] please help me How i can solve this question because I don't understand it clearly

Hey everyone, I'm new to the forums so sorry if I'm making a post that's obviously ridiculous but my level of understanding only takes me so far, hence the need to post here. I just finished my bachelor of science degree in Biochemistry this summer but only have an A-level qualification in...

Hello all, May someone help me on this question: Suppose the map F is an isometry which maps a dense set H of a semi-normed space \mathcal{H} to a normed space \mathcal{G} , now the theorem said we can extend this isometry in a unique manner to a linear isometry of the semi-normed...

Pretty much knows the triangle inequality. \left| a + b \right| \le \left| a \right| + \left| b \right| I was reading a source which asserted the following extension of the triangle inequality: \left| a + b \right|^p \le 2^p \left(\left| a \right|^p + \left| b \right|^p\right) This is...

Homework Statement Define f(6) in a way that extends to be continuous at s=6. Homework Equations None. Only limits are required. The Attempt at a Solution In order to figure out which point needs to be added to the function, I have to find the limit of this function as s->6. This will...

Hello everyone, I need some help with finding norms of the field extension. I feel pretty comfortable when representing norms as determinants of linear operators but I seem to be stuck with representing norms as product of isomorphims. I have read Lang's GTM Algebra, but I really would...

Homework Statement Let f : A --> Y be a continuous function, where A is a subset of X and Y is Haussdorf. Show that, if f can be extended to a continuous function g : Cl(A) --> Y, then g is uniquely determined by f. The Attempt at a Solution I think I can solve this on my own, but I...

Here's an extension of a list posted earlier. If anybody can think of any additions to the list, please post :D! Perspectives of the world: ------------------------------- Optimist – The glass is half-full. Pessimist – The glass is half-empty. Existentialist – The glass is. Fatalist – The...

normal extension - an algebraic field extension L/K is said to be normal if L is the splitting field of a family of polynomials in K[X]. (wikipedia) I have a question about the 'family of polynomials' it says the family should be arbitarary large? if E be an algebraic closure of Q(rational...

hey all, this is my first post, sorry for not introducing myself formally, but i am on a tight schedule, getting ready for my retakes. my question is; a 5kg monkey, initially at rest, starts climbing up the weightless rope at 0.2g. the top end of the spring is connected to a spring with...

How can the function f: ℝ² → ℝ : (x,y) |--> {{x^2+y^2-(x^3y^3)}\over{x^2+y^2}} if (x,y) ≠ (0,0) be defined in the origin so that we get a continuous function?When I take 'x=y' (so (y,y)) and 'y=x' (so (x,x)) I get: {{2-y^4}\over{2}} and {{2-x^4}\over{2}} So for the first one I get '1'...

We have a trebuchet which we can adjust the launch angle, and how far the rubber band is pulled back (extension). The rubber band spring constant is constant throughout. I need to find the relationship between the angle of launch and range, and the relationship between the extension and range...

Given metric spaces (X, d_X), (Y, d_Y), and subsets \{ x_1, ..., x_n \}, \{ y_1, ..., y_n \} of X and Y respectively, if I define a function that send x_i to y_i, what are the minimal restrictions we need on X and Y in order for there to exist an extension of that map to a continuous...

Based on the results of every variation of the two-slit experiment so far, the presence or absence of the interference pattern is based on whether or not which-path information can be known or not. The Delayed Choice Quantum Eraser experiment...

Homework Statement "Given a function H, assigning a value, 0 or 1, to each atomic sentence, define a sequence ℑ0, ℑ1, ℑ2, ℑ3,... of functions, as follows: ℑ0 is just H. Given a function ℑn, assigning a value, either 0 or 1, to the sentences of degree less than or equal to n, define the...

Homework Statement A cylindrical steel bar, 20-mm in diameter and 5.0-m in length, is subjected to an axial tensile load of 40-kN applied at the ends of the steel bar. Determine the axial extension, in millimeters, of the steel bar under this loading condition?Homework Equations E=\sigma/e...

Hi... How do I construct a Galois extension E of Q(set of rational numbers) such that Gal[E,Q] is isomorphic to Z/3Z. Thanks.

Homework Statement A wire of diameter 5.0 mm supports a 2.8kg load. (a) Determine the tension in the wire (b) The original length of the wire was 2.0m Calculate its extension when supporting the load. Homework Equations Young's modulus for the material of the wire = 2.0 x 10^7N...

I am having no luck understanding how to find the basis of a field adjoined with an element. For example Q(sqrt(2)+sqrt(3)) I know that if i take a=sqrt(2)+sqrt(3) that i can find a polynomial (1/4)x^4 - (5/2)x^2 + 1/4 that when evaluated at a is equal to zero. So, from that I know the...

For wave equation: \frac{\partial^2 u}{\partial t^2} \;=\; c^2\frac{\partial^2 u}{\partial x^2} \;\;,\;\; u(x,0)\; =\; f(x) \;\;,\;\; \frac{\partial u}{\partial t}(x,0) \;=\; g(x) D'Alembert Mothod: u(x,t)\; = \;\frac{1}{2} f(x\;-\;ct)\; +\; \frac{1}{2} f(x\;+\;ct)\; +\; \frac{1}{2c}...

First, if there are tests that have ruled one way or the other, that pretty much ends this discussion. I couldn't find any, but if they exist please point them out here. That leaves the discussion of: 1) How could these been distinguished experimentally, and 2) How precise do tests have...

This stuff is killing me... Let K \leq M \leq L be fields such that L is galois over M and M is galois over K. We can extend \phi \in G(M/K) to an automorphism of L to show L is galois over K. I need help filling in the details in why exactly L is galois over K.