Splitting Definition and 259 Threads

Homework Statement Describe the splitting field of x^9-1 over Z_13. Homework EquationsThe Attempt at a Solution Well, seeing if any other of {1,...,12} would be some tricky arithmetic. Is my best bet here moving forward to just divide x^9-1 by x-1 since I know 1 is a root and then go from...

Homework Statement I need to construct a splitting field of f(x)=x^4-x^3-5x+5 over Q Homework EquationsThe Attempt at a Solution So first I will assume r is a root and divide f(x) by (x-r). The quotient came out to be x^3 + (r-1)x^2 + (r^2-r)x + r^3 - r^2 - 5. I am a bit confused what to do...

Homework Statement I've been given the worked answer to a very similar question, but there's a step I don't understand so I can't apply it. My question asks: An energy level with ##J=\frac{11}{2}## has six hyperfine sub-levels with these relative energies...

Homework Statement Hello PF. I need to find a splitting field of x^4-7x in C over Q Homework EquationsThe Attempt at a Solution letting r be a root, I did the division and got x^4-7x = (x-r)(x^3+r*x^2+x*r^2+r^3). I'm a little confused on what to do now, do I just take another root and do the...

Homework Statement Construct a splitting s for the polynomial x^3+2x+1 over Z/Z3 Homework Equations 4=1 Mod 3 :P The Attempt at a Solution So I'm actually quite confused. There are no roots for x+3+2x+1 over Z/Z3. I am used to constructing splitting fields with polynomials that have...

Hey! :o Let $F\subseteq E\subseteq K$ be consecutive field extensions and $f\in F[x]$ be non-constant. I want to show that if $K$ is the splitting field of $f$ over $F$, then $K$ is the splitting field of $f$ also over $E$. Since $K$ is the splitting field of $f$ over $F$, we have that $f(x)...

Hey! :o Let $F$ be a field, $f\in F[x]$ be non-constant and $K$ the splitting field of $f$ over $F$. Let $g\in F[x]$ be a non-constant polynomial, that divides $f$. I want to show that $g$ can be splitted into linear factors of $K[x]$. Is $K$ the splitting field of $g$ over $F$ ? Since...

I recently decided to take a whack at this problem. Came up with an interesting approach, thought it would make a good conversation topic. Anyone else tried to do this? What were your results?

There is several things I don't quite understand about crystal-field splitting. Q1: Why is crystal-field effect is not explained for f-orbitals like in d-orbitals in terms of real-orbitals? Correct me if I am wrong, but typically for d-orbitals, crystal-field splitting is explained classically...

What is the cause of hyperfine line splitting? I saw somewhere that it was because of spin quantum number and somewhere else that it was because of how electrons interact in the atom. And relative intensity of spectral lines? I'd assume it's due to the probability of electron jumps being more...

A large number (56) of 18 gauge-ish connection pins are presented by the connector. They are wired together in large groups on the "client" side of the connector (4 electrodes). We wish to re wire these groups together and then split them off into 24x 18 AWG a few feet removed from the...

Let f(x)=x4-2x2+9 Find the splitting field and Galois group for f(x) over ℚ Here is what I have written out so far. If I have found the splitting field E correctly, have I proceeded with the Gal(E/F) group correctly? Also, how would I go about finding the roots of this equation by hand...

Hi, I am trying to understand the meaning in two level system of splitting of the states due to high light intensity. the main question that I asked my self that if I had an atomic vapor and I excited the vapor in high intensity laser beam the levels will be split and I will observe Mollow...

Hello everyone, Does anyone know much about the crystal field splitting of the f orbitals in actinides, specifically uranium? I am doing research on the chemistry of uranium and have come across an interesting problem. U(IV) compounds undergo a transition to a non-magnetic singlet ground state...

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...

Homework Statement Split the function f(x) = ex + πe−x into odd and even parts, and express your result in terms of cosh x and sinh x. Homework Equations f(x) = 0.5[f(x) + f(-x)] +0.5[f(x) - f(-x)] The Attempt at a Solution So i know that: ex = 0.5[ex - e-x] + 0.5[ex + e-x] = sinh(x) +...

I have a hammock that is rated to 400 lbs. I want to use two neodymium magnets to suspend the hammock from two steel posts. Do I need two magnets rated 400 lbs each?

…… is the splitting of light into its constituent colors (wavelengths). Can I use the term ‘dispersion’ here? Or is ‘dispersion’ only restricted to splitting due to phase velocity/refraction differences of different wavelengths (prism)? If not ‘dispersion’, is there another general term that...

Homework Statement [/B] Hi there! This is a question from a practice problem sheet I got from the lecturer of my Condensed Matter 1 course. Below are the normal vectors to the {111} and {112} lattice planes: Homework Equations [/B] Bragg Condition: \begin{equation} n \lambda = d \sin...

After doing a couple courses in physics as well as calculus and differential equations, I was starting to wonder about splitting a derivate, such as ## \frac{dy}{dx} ##, into separate pieces ##dy## and ##dx##. I know we've never done it in calculus or differential equations because it isn't...

How are power distribution cables parallel-ed to split loads optimally? If the cables were exactly identical this would be easy but how is it achieved in general? e.g. Say there's a 11 kV to 420 V distribution transformer but the primary load is some distance away. Assume the load increases...

I recently saw a phenomenon wherein a metal droplet was stretched and stuck between two electrodes that it wet to. Upon applying electric field, the droplet split into two. ANy idea why?

HI all, I think I have to use some combination of waveplates on the signal, but I am not sure, does anybody know this? Thanks for help

I am reading through my course guide and problems worked out and something bothers me. I think at this point we are working with classical waves, moving towards QM eventually. The idealization we work with is this. We have a light source creating a wave ψ(x,t)=Acos(ωt+kx) . The light beam...

Is there a way to split gamma-ray photons into less energetic photons? And is the opposite possible, by combining low energy photons into a high energy one?

Homework Statement Evaluate Integrate (2-3x/(Sqrt.(1 - x^2))) dx Homework Equations 1/Sqrt.(1-x^2) = arctan The Attempt at a Solution I am so lost, but this is what I've tried, but didn't work... I separated the integral into two so Integral of (2/(Sqrt.(1-x^20))) dx - integral of...

How did they arrive at this expression? I understand the ##Q^2## term in the nominator and the ##r = r_0 (A)^{\frac{1}{3}}## term in the denominator. Where did the ##\frac{\alpha}{2}## term come from? Putting in Z=40, A=100 gives ##3.3 \times 10^{14} J##, not ##65 MeV##. The energy term...

I have read that Lyman series of Hydrogen atom occur on the n = 2 level, and it is due to electron orbital coupling. Suppose I understand what is an electron-orbital coupling, and that one gets 3/2 and 1/2 total angular momentum for n = 2 level. But, from quantum mechanical derivation of the...

My textbook is going through an example on splitting fields. It asked to find a splitting field for x^4 - 6x^2 - 7 over the rational numbers. This polynomial factors to (x^2 - 7)*(x^2+1) which has roots of 7^(1/2) and i. So i figured the extension field E we are looking for is Q(i)(7^(1/2)), but...

Hey, I've been trying to figure out why a protons in identical environments will not undergo spin-spin splitting with one another. From what I understand splitting of a signal for a proton is due to the nucleus of another nearby either opposing the external magnetic field or acting in the same...

1. The problem statement, all variafbles and given/known data Problem: Let E be a splitting field of f over F. If [E:F] is prime, show that E=F(u) for some u in E (show that E is a simple extension of F) Homework Equations Things that might be useful: If E>K>F are fields, where K and F are...

Homework Statement (a)Find splitting between F=0 and F=1 in hydrogen (b) Find the constant ##A## and nuclear magnetic moment Homework EquationsThe Attempt at a Solution Hyperfine splitting is given by: H_{hf} = \frac{A}{2} \left[ F(F+1) - I(I+1) - J(J+1) \right] The intervals are given by...

Hello :o The extension $\mathbb{Z}_p \leq \mathbb{F}_{p^n}$ is normal, as the splitting field of the polynomial $f(x)=x^{p^n}-x$ ($\mathbb{Z}_p$ is a perfect field therefore each polynomial is separable). So, if $a \in \mathbb{F}_{p^n}$, then $q(x)=Irr(a,\mathbb{Z}_p)$ can be splitted over...

I saw recently a new article about some kind of PEC cell immersed in water which could separate water into hydrogen and oxygen. (It was news because they found a way to protect the cell from immediate corrosion damage) It seemed that somehow the scientists had designed a cell that could put the...

How does one prove the following relation? \int_{a}^{b}f(x)dx= \int_{a}^{c}f(x)dx + \int_{c}^{b}f(x)dx Initially, I attempted to do this by writing the definite integral as the limit of a Riemann sum, i.e. \int_{a}^{b}f(x)dx=...

Homework Statement Sketch d-orbital splitting diagrams for a complex in Td symmetry and a complex in D4h symmetry. Label the orbitals as bonding, antibonding, or non-bonding Homework EquationsThe Attempt at a Solution So I know that in Td symmetry, the d-orbitals split into a lower...

Hey all, This is a problem I can't really wrap my head around. 1.) The lanthanoid series is formed through the progressive filling of the 4f subshell, whereas the first row transition metal series is formed through the progressive filling of the 3d subshell. a) What is the relative...

Hi, so i have finished this question but I am not sure if I did it right. Could you guys check it and tell me if its right or show me where i went wrong. This is the question: A space vehicle traveling At a velocity of 1400m/s separates by a controlled explosion into two sections of mass 859kg...

I just found out about this via Twitter: http://phys.org/news/2014-10-function-electron.html I'm too tired to have got my head around all the details, but it looks as if there's a fascinating new experimental perspective on what a "measurement" in QM actually is. DOI for the original journal...

Hey! :o Could you give me some hints how to find a splitting field for the polynomial $$x^{p^{80}}-1$$ where $p$ is prime?? (Wondering)

Consider the above experimental setup. Here the source can emit one photon at a time. The output from the source is sent to a beam splitter which divides it into three possible paths of equal probability. Splitter 2 further splits the path 3 into two paths of equal probability. We have...

Forgive me if this is a foolish question but, what caused the super force in the early universe to split into the weak and strong forces and obviously without them the universe as we know it wouldn't exsist.

I have some XPS spectrums that I am trying to fit (my first time doing so), using XPSpeak. I understand that for spin-orbit splitting the FWHM, line shape (i.e.% gaussian/lorentzian) must be equal (more or less), peak area ratios set (i.e. 2:3 for 3d3/2 and 3d5/2), and the peak separation...

I am reading Beachy and Blair's book: Abstract Algebra (3rd Edition) and am currently studying Theorem 6.5.2. I need help with the proof of the Theorem. Theorem 6.5.2 and its proof read as follows:In the conclusion of the proof, Beachy and Blair write the following: " ... ... Hence, since F...

I am reading Section 6.4: Splitting Fields in Beachy and Blair: Abstract Algebra. I am currently studying Example 6.4.2 on page 290 which concerns the splitting field of x^3 - 2 \text{ over } \mathbb{Q} . In Example 6.4.2, B&B show that the splitting field of x^3 - 2 \text{ over }...

I am struggling to understand how to determine splitting patterns for molecules. For example, (see attached) I was trying to determine the splitting pattern for CH2 in the molecule. On the right i believe n is equal to 1 and on the right it is equal to 3, but i am not sure if this is correct. A...

Homework Statement Estimate the magnitude of the fine structure splitting in H-α in THzHomework Equations Rydberg -- R_y \left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right) = \Delta E The Attempt at a Solution This isn't really a request for solution help, and more a justification. I know that if...

Homework Statement The interaction between the spins of the two particles in a hydrogenic atom can be described by the interaction Hamiltonian $$\hat{H_I} = A \hat{S_1} \cdot \hat{S_2}.$$ Compute the splitting of the ground state due to ##\hat{H_I}##. Both particles have spin 1/2. Homework...

I am reading Dummit and Foote Section 10.5 Exact Sequences - Projective, Injective and Flat Modules. I need some help in understanding D&F's proof of Proposition 25, Section 10.5 (page 384) concerning split sequences. Proposition 25 and its proof are as follows...

Homework Statement Investigate The number of atoms splitting up in a nuclear reaction and Bacterial growth Your investigation should use appropriate formulae and graphs and include information from records of real events. You should aim to show how good the mathematical models are in...