Extension Definition and 294 Threads

Homework Statement Consider the field extension Q(c):Q, where c is a primitive nth root of unity. Is this extension normal? Homework EquationsThe Attempt at a Solution I believe this polynomial splits in x^n - 1,where it's n roots are exactly the powers of c. Thus this extension is normal...

Hey! :o Let $E/F$ be a finite extension. I want to show that this extension is Galois if and only if $E$ is a splitting field of a separable polynomial of $F[x]$. I have done the folllowing: $\Rightarrow$ : We suppose that $E/F$ is Galois. So, we have that the extension is normal and...

Hey! :o Let $E/F$ be a finite Galois extension and let the chain of extensions $F = K_0 \leq K_1 \leq \dots \leq K_n = E$. Let $G = Gal(E/F)$ and, for $i = 0, 1, \dots , n$, let $H_i$ be the subgroup of $G$, that corresponds to $K_i$ through the Galois mapping. I want to show that, for any...

Homework Statement let b be a square root of 1+i, show that Q(b):Q is not a normal extension. Also, what is the Galois group of the extension? Homework EquationsThe Attempt at a Solution so b = +/- (1+i)^(1/2), and it's minimal polynomial is x^4+4 which has roots -(2)^1/2 and 2^(1/2) that are...

Homework Statement Let [ S] = {2^(1/n) | for all n in the natural numbers}, is Q[ S] algebraic? finite? simple? separable? Homework EquationsThe Attempt at a Solution I believe it is algebraic because every element of [ S] will be a root of x^n-2, and every element of Q is obviously algebraic...

does this paper Standard Model-Axion-Seesaw-Higgs Portal Inflation. Five problems of particle physics and cosmology solved in one stroke Guillermo Ballesteros, Javier Redondo, Andreas Ringwald, Carlos Tamarit (Submitted on 5 Oct 2016) We present a minimal extension of the Standard Model (SM)...

Hey! :o Let $E/F$ be a Galois extension. I want to show the following: $F\leq K\leq E \Rightarrow \mathcal{F}(\mathcal{G}(E/K))\geq K$ $H\leq \mathcal{G}(E/F)\Rightarrow \mathcal{G}(E/\mathcal{F}(H))\geq H$ Since $E/F$ is a Galois extension, we have that the extension is normal and...

Young's Modulus is usually defined as the intrinsic property of a material indicating it's stiffness, or it's ability to resist deformation. Though, it is measured in Pa, meaning it should have some statistical description. Spring constant, for example, can be define as the stiffness of an item...

Hey! :o I want to show that each extension of degree $2$ is normal. I have done the following: Let $K/F$ the field extension with $[F:K]=2$. Let $a\in K\setminus F$. Then we have that $F\leq F(a)\leq K$. We have that $[K:F]=2\Rightarrow [K:F(a)][F(a):F]=2$. There are the following...

Homework Statement Let c be a primitive 3rd root of unity in the complex numbers and b be the real root of x^4-2=0. If a = c*b, show that Q(b,c) = Q(a)Homework EquationsThe Attempt at a Solution So [Q(a):Q(c)]=3 and [Q(a):Q(b)]=4, and c and b contain no 'overlapping material', so [Q(a):Q)=12...

Hey! :o Let $f = x^4−2x^2−1 \in \mathbb{Q}[x]$. We have that $f(x+1)=(x+1)^4-2(x+1)^2-1=x^4+4x^3+6x^2+4x+1-2(x^2+2x+1)-1=x^4+4x^3+4x^2-2$ We have that $p=2$ divides all the coefficients $4,4,-2$ and $p^2=4$ does not divide the constant term $-2$. So, the polynomial $f(x+1)$ is Eisenstein...

Question. Is it true that a finite extension $K:F$ is simple iff the purely inseprable closure is simple over $F$? I think have an argument to support the above. First we show the following: Lemma. Let $K:F$ be a finite extension and $S$ and $I$ be the separable and purely inseparable...

Homework Statement If a question gives us a max load with different extensions, how do I calculate the strain? For example: 1900KN=x mm 1900KN=x mm 2000KN=x mm 2000KN=x mm 2000KN=x mm 1900KN=x mm 1900KN=x mm Homework Equations strain= ΔL / L The Attempt at a Solution What I think is...

So, complete hypothetical on laws governing power transmission over a distance: Lets say I have a generator as a source of power - and attached to that is FIVE KILOMETERS worth of extension cords. :) In this situation, what laws of electricity do I need to look out for? Voltage drop over...

Hi, I have a friend with whom I got into an argument of what happens when you multiple power cord extension connected. So here is the problem: - There are 3 extension power cords, each power cord has 3 plugs. - First extension power cord is connected to the wall outlet - Second one is...

Homework Statement Let q be a root of p(x) = x^3 + x^2 + 1 in an extention field of Z2 (integers modulus 2). Show that Z2(q) is a splitting field of p(x by finding the other roots of p(x) hint: this question can be greatly simplified by using the frobenius automorphism to find these zero's...

Homework Statement A spring with spring constant 50 N/m is pulled with 10 N force at both ends of the spring. So, the extension of spring length is.. A. 0.0 m B. 0.1 m C. 0.2 m D. 0.3 m E. 0.4 m Homework Equations F = k Δx The Attempt at a Solution F = 10 N <--- 0000000000000000000000...

I am reading Dummit and Foote's book: Abstract Algebra ... ... and am currently focused on Section 10.4 Tensor Products of Modules ... ... I have a basic question regarding the extension of the scalars ... Dummit and Foote's exposition regarding extension of the scalars reads as...

I am reading Dummit and Foote's book: Abstract Algebra ... ... and am currently focused on Section 10.4 Tensor Products of Modules ... ... I have a basic question regarding the extension of the scalars ... Dummit and Foote's (D&Fs) exposition regarding extension of the scalars reads as...

Hi. I am reading a paper about gaussian beams and the author says that gaussian beams have simultaneously minimal divergence and minimal transversal extension. In order to prove it, the author states that \mathrm{divergenece} \propto \int_{-\infty}^{+\infty} \frac{d\,k_{x}}{2\pi}...

Hello! I took a quality certification test yesterday and there was a question on there about torque wrench formulas. I didn't have anything about that in my notes, so I took some time this morning to create a cheat sheet based on what I could find on the web. 1) In some of the forums, they...

Homework Statement A block of mass 1 kg is attached to a spring. The spring extends by 10 cm. Find spring constant.Homework EquationsThe Attempt at a Solution Potential energy of spring = kx2/2 work done by block = PE Hence mg*x=kx2/2 ∴1*9.8*0.1=k*0.1*0.1/2 ∴k=196N/m But solution says mg = kx...

This is a question that came about while I attempting to prove that a simple extension was a splitting field via mutual containment. This isn't actually the problem, however, it seems like the argument I'm using shouldn't be exclusive to my problem. Here is my attempt at convincing myself that...

I am happy with my solutions of questions 1-4 below, but need some help on question 5. 5. The squares in the reals are simply the positive reals and the non-squares are the negative reals so the quotient of two no squares is the quotient of two negative reals that is a positive real of in...

Homework Statement A block of mass ##M = 1 kg## is placed of a fixed rough incline of inclination ##\theta=sin^{-1} \frac{7}{10}## and coefficient of friction ##\mu=\frac{1}{\sqrt{51}}##. It is connected to a spring of spring constant 100 N/m. Initially the spring is in natural state with...

I was reading the Wikipedia page on Dynamism in order to get an idea of the motivation and thinking behind Liebniz's physics. In it there is this paragraph: In the opening paragraph of Specimen dynamicum (1692), Leibniz begins by clarifying his intention to supersede the Cartesian account of...

Homework Statement (a) An 81 kg student is launched from a bridge by his best friends, some 50 metres above the river surface. Fortunately, he is attached to a 30 m bungee cord with a spring constant of 270 N/m. i) What is the equilibrium length of the bungee cord, including the force of...

Homework Statement Two identical disks with mass m and radius r are connected via a massless wire of length L which is winded up around both disks. Disk B is connected to the ceiling and is free to rotate around its axis. Disk A is besides disk B and will fall due to the gravitational force...

$\newcommand{\R}{\mathbf R}\newcommand{\C}{\mathbf C}$ Low-Tech Complexification: Let $V$ be a finite dimensional vector space over $\R$. We can forcefully make $W:=V\times V$ into a complex vector space by defining addition component-wise and product $\C\times W\to W$ as $$ (a+ib)(u...

Homework Statement In the figure shown below all surfaces are friction-less. Find the maximum extension in the spring(in meters) , if the blocks are initially at rest and the spring is initially in its natural length. Details and Assumptions: F=30N k=700N/m m=5kgHomework EquationsThe Attempt...

Hello everyone, I graduated with a BSEE in 2008 and tried a PhD program in EE (semicon device specialization) for a while but it didn't work out. I struggled with an illness that forced me to drop out. I have been working in industry (NOT The semicon industry) since leaving the PhD program...

So I have the metric as ##ds^{2}=-(1-\frac{2m}{r})dt^{2}+(1-\frac{2m}{r})^{-1}dr^{2}+r^{2}d\Omega^{2}##* I have transformed to coordinate system ##u,r,\phi, \theta ##, where ##u=t-r*##(2), where ##r*=r+2m In(\frac{r}{2m}-1)## and to the coordinate system ##v,r,\phi, \theta ##, where...

The coordinates ##u## and ##v## are defined as ##u=t+r*##, ##v=t-r*##, where ##r*=r+2M In(\frac{r}{2M}-1)##. In ##u,r,\theta,\phi ## coordinates the radially null geodesics are given by: ##\frac{du}{dr}=0 ## for infalling, ##\frac{du}{dr}=2(1-\frac{2M}{r})^{-1} ## for outgoing. In the...

Suppose ##\mu:\mathcal{F}\rightarrow[0,\infty)## be a countable additive measure on a ##\sigma##-algebra ##\mathcal{F}## over a set ##\Omega##. Take any ##E\subseteq \Omega##. Let ##\mathcal{F}_{E}:=\sigma(\mathcal{F}\cup\{E\})##. Then, PROVE there is a countable additive measure...

Hello everyone, I have designed a part that should be able to withstand a shear stress of 43 MPa ( give or take). The part will be made of Nickel Aluminum Bronze (CuAl10Ni5Fe4). When I check yield strength values for this material (I'll multiply it by 0.6 to estimate its shear strength), they...

Homework Statement A 50g block is attached to a vertical spring whose stiffness constant is 9N/m. The block is released at the position where the spring is unextended. What is the maximum extension of the spring? How long does it take the block to reach the lowest point? Homework Equations...

In Hooke's law, F=-kx. Assume that a mass is hung from the end of an ideal massless spring, the spring stretches a distance of x. However, in real life the spring has mass and it is submerged in a "fluid" of air. Compared to an ideal spring, would the real spring have a slightly different...

Homework Statement hi guys i have a sheet of i have the W,L and T a force and also the elastic modules etc now my issue is i need to find the strain i have the stress etc but the is no extension figure given i have the usual strain equation e=x/l can anyone point me in the right direction in...

My textbook says: If u is transcnedetal over F, it is routine to verify that: F(u) = {f(u)g(u)^(-1) | f,g in F[x]; g /= 0} However, me being the scrubbiest of all scrubs does not understand what they did here. First of all, I don't understand why they needed to invert the g(u) function. It...

Ive been given a question which I'm stuck on and cannot answer. I've only been able to calculate the maximum allowable working stress and other then that I am stuck on how to answer the following question: A specimen of the same material that was used in the above test (mild steel), had a cross...

I'm trying to solve this exercise but I have some problems, because I haven't seen an exercise of this type before. "f(x)= \pi -x in [0, \pi] Let's consider the even extension of f(x) in [-\pi, \pi] and write the Fourier Series using this set ( \frac{1}{\sqrt{2 \pi}}, \frac{1}{\sqrt {\pi}}...

Homework Statement see attachement Homework Equations dl= PL/AE The Attempt at a Solution as you see here my youngs modulus E varies linearly. i doubt that adding up E1 &E2 wld give solution.. any help?

Hello, i am relatively new linux user so please be as thoroughly descriptive as you can. i am starting a project and i have a file "file.hdf5" which contains velocity data i need to process in my fortran program. What do i have to do in order to use the data inside this file? i normally use...

Hey! :o Let $p$ be an odd prime number. We set $a=Re \left (e^{\frac{2\pi i}{p}}\right )$. Show that: 1) $$\mathbb{Q}(a)\leq \mathbb{Q}(e^{\frac{2\pi i}{p}}) \text{ and } \left [\mathbb{Q} \left (e^{\frac{2\pi i}{p}} \right ):\mathbb{Q}(a)\right ]=2$$ 2)...

Good Evening! I have a few files with "mlp" extension, which are pH vs Time graphs from a lab data logger. The original software is unavailable to me, but in the Internet I can`t find anything to open it. Please, help!

Homework Statement A weight of ##50## N is suspended from a spring of stiffness ##4000## N/m and is subjected to a harmonic force of amplitude ##60## N and frequency ##6## Hz. Homework EquationsThe Attempt at a Solution Since ##W = mg = 50##, we have that the suspended mass, ##m = 5.10204##...

A weight of \(50\) N is suspended from a spring of stiffness \(4000\) N/m and is subjected to a harmonic force of amplitude \(60\) N and frequency \(6\) Hz. Since \(W = mg = 50\), we have that the mass, \(m = 5.10204\), and we know that \(f = \frac{\omega}{2\pi} = 6\) so \(\omega = 12\pi\)...

I have an idea about an extension cord that is flat and sticky like tape. This will cause less mess around the TV and will prevent anyone from tripping over any cords

I just watched a video discussion on the modern interpretations of the wave function. In it I was introduced to QBism, i.e. Quantum Bayesianism. To me sounded a lot like the famous Monty Hall problem. Is QBism's probability similar to that?

Hello! From a data set of F-x measurements of a single dsDNA molecule I want to calculate the persistence length P . So I plotted \frac {1} {\sqrt{(F)}} vs. x and fitted these data points (linear). According to an interpolation formula the extension x of a worm like chain with contour...