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Math is easy (by Charles Daney)
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More concepts from ring theory | Mathophilia
https://mathophilia.wordpress.com/2015/05/05/more-concepts-from-ring-theory
Why do we care about unique factorization? Factorization of prime ideals in extension fields →. More concepts from ring theory. May 5, 2015. We are at an important juncture in our discussion of algebraic number theory. From here on out, the path starts to go uphill more steeply, with quite a bit more abstraction and technical complexity. I hope you’ll follow along anyhow. Don’t feel like you need to grasp all the fine points immediately. The previous discussion of rings is here. Ab = a ⋅ b for all a...
Groups and rings | Mathophilia
https://mathophilia.wordpress.com/2015/05/05/groups-and-rings
Fields and Galois theory →. May 5, 2015. In our previous installment of the series on algebraic number theory, we took a little detour into Diophantine equations. In order to provide some motivation for the theory itself. Prior to that, we had looked at different types of numbers. To give a perspective on the sorts of objects the theory deals with. In those discussions, we touched (explicitly or implicitly) on abstract algebraic structures called groups. G is a mathematical system consisting of a set of ...
Cyclotomic fields, part 2 | Mathophilia
https://mathophilia.wordpress.com/2015/05/05/cyclotomic-fields-part-2
Roots of unity and cyclotomic fields. Algebraic number theory – index →. Cyclotomic fields, part 2. May 5, 2015. In our previous article. On cyclotomic fields we were talking about why the Galois group G of ℚ(μ. 8474; is isomorphic to (ℤ/nℤ). Where n∈ℤ and μ. Is the group of n. Roots of unity, the roots of x. 1=0 in some extension of ℚ. (Check here. For a list of previous articles on algebraic number theory.). So far we’ve shown that G is isomorphic to a subgroup of (ℤ/nℤ). The group μ. Is a primitive n.
Roots of unity and cyclotomic fields | Mathophilia
https://mathophilia.wordpress.com/2015/05/05/roots-of-unity-and-cyclotomic-fields
Splitting of prime ideals in algebraic number fields. Cyclotomic fields, part 2 →. Roots of unity and cyclotomic fields. May 5, 2015. In preparation for many good things that are to come, we need to have a talk about another important class of field extensions of ℚ – the cyclotomic extensions. (Check here. For a list of previous articles on algebraic number theory.). Since f(1)=0, x-1 is one factor of f(x), and f(x)/(x-1) = x. Roots of unity – denoted by ζ. For some integer k, 1≤k n. So the smalle...
Modular arithmetic | Mathophilia
https://mathophilia.wordpress.com/2015/05/05/modular-arithmetic
Fields and Galois theory. Rings: fields of fractions →. May 5, 2015. After some hiatus, let’s return to our discussion of algebraic number theory. Check here. We have already talked briefly about groups and rings ( here. But because ring theory is so important to the whole subject, we need to go into it a lot deeper. As preparation for that, we’ll look at some of the simplest examples, which will turn out to be greatly generalized in the sequel. In any case, the construction is so important and pervasive...
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Particle Decelerator: The Beam of Darkness
http://decelerator.blogspot.com/2013/12/the-beam-of-darkness.html
Slowing down particles of quantum and cosmological data. Sunday, 15 December 2013. The Beam of Darkness. This week, engineers from Singapore. Released details of a fascinating prototype which takes a radically different approach to the creation of an invisibility cloak. Whilst still popularly considered the realm of science fiction, many working prototypes for 'invisibility cloaks' have been demonstrated. In the past few years, most using metamaterials. And some using tiny antennae. In their full paper.
Particle Decelerator: June 2013
http://decelerator.blogspot.com/2013_06_01_archive.html
Slowing down particles of quantum and cosmological data. Friday, 21 June 2013. Plants doing quantum physics. This week the fledgling field of quantum biology. Took a major step forward with the publication of a significant study in Science. In the study, scientist, Niek van Hulst, and colleagues at the Institute of Photonic Sciences. As Jason Palmer writes. Has for some years now been suggesting a far deeper entanglement between large complex biological organisms and quantum mechanics. Various resear...
Particle Decelerator: First Direct Evidence of Cosmic Inflation
http://decelerator.blogspot.com/2014/03/first-direct-evidence-of-cosmic.html
Slowing down particles of quantum and cosmological data. Monday, 17 March 2014. First Direct Evidence of Cosmic Inflation. Swirls in the CMB polarization, shown here in the BICEP2 data, show the first clear evidence of primordial gravitational waves. Image: BICEP2. The title of the scientific presentation said it all:. First Direct Evidence of Cosmic Inflation". It appears that the rumours were true. Has discovered something extraordinary. The FAQ summarising the data. No experiment should be taken too s...
Particle Decelerator: There, but invisible
http://decelerator.blogspot.com/2013/12/there-but-invisible.html
Slowing down particles of quantum and cosmological data. Tuesday, 31 December 2013. There, but invisible. By Anthony McCall. Commissioned by AV Festival 2006. In physics, the darkness is the most illuminating place to look, if you want to understand the Universe right now. This year, Planck. Told us that 26.8% of all matter is dark. And that the energy responsible for that - dark energy. Accounts for 68.3% of all energy. But what this means is that we live in some of the most exciting times imaginable.
Particle Decelerator: Primordial Gravitational Waves and Cosmic Inflation
http://decelerator.blogspot.com/2014/03/primordial-gravitational-waves-and.html
Slowing down particles of quantum and cosmological data. Monday, 17 March 2014. Primordial Gravitational Waves and Cosmic Inflation. Today might turn out to be a historic day for the field of cosmology. At 16:00 UTC at the at Harvard-Smithsonian Center for Astrophysics, physicists will be announcing. What they are describing as a "major discovery". Since the news of the announcement broke last week, the cosmology community has been awash with rumours as to what the discovery might be. Posited by Alan Guth.
Particle Decelerator: January 2013
http://decelerator.blogspot.com/2013_01_01_archive.html
Slowing down particles of quantum and cosmological data. Sunday, 20 January 2013. Can particle physics transform our online experience? One of the more intriguing papers we've happened upon early this year is some research from the University of Fribourg. Which suggests that particle physics can improve the technology behind recommendation systems. But what on earth does particle physics have to do with software like that? Well, a group of researchers at the at the University of Fribourg in Switzerland.
Particle Decelerator: March 2014
http://decelerator.blogspot.com/2014_03_01_archive.html
Slowing down particles of quantum and cosmological data. Monday, 17 March 2014. First Direct Evidence of Cosmic Inflation. Swirls in the CMB polarization, shown here in the BICEP2 data, show the first clear evidence of primordial gravitational waves. Image: BICEP2. The title of the scientific presentation said it all:. First Direct Evidence of Cosmic Inflation". It appears that the rumours were true. Has discovered something extraordinary. The FAQ summarising the data. No experiment should be taken too s...
Particle Decelerator: October 2013
http://decelerator.blogspot.com/2013_10_01_archive.html
Slowing down particles of quantum and cosmological data. Tuesday, 8 October 2013. Here, at last . Here at last .". With those words the Nobel Committee awarded the 2013 Nobel Prize for Physics. To Francois Englert (left) and Peter Higgs (right) for their work in discovering the theoretical basis for what we now refer to as the Higgs mechanism. Speculation about the 2013 prize had reached fever pitch. Ahead of this year's announcement. Most commentators in the physics community. When Joe Incandela and Fab...
Particle Decelerator: Here, at last ...
http://decelerator.blogspot.com/2013/10/here-at-last.html
Slowing down particles of quantum and cosmological data. Tuesday, 8 October 2013. Here, at last . Here at last .". With those words the Nobel Committee awarded the 2013 Nobel Prize for Physics. To Francois Englert (left) and Peter Higgs (right) for their work in discovering the theoretical basis for what we now refer to as the Higgs mechanism. Speculation about the 2013 prize had reached fever pitch. Ahead of this year's announcement. Most commentators in the physics community. When Joe Incandela and Fab...
Particle Decelerator: Plants doing quantum physics
http://decelerator.blogspot.com/2013/06/plants-doing-quantum-physics.html
Slowing down particles of quantum and cosmological data. Friday, 21 June 2013. Plants doing quantum physics. This week the fledgling field of quantum biology. Took a major step forward with the publication of a significant study in Science. In the study, scientist, Niek van Hulst, and colleagues at the Institute of Photonic Sciences. As Jason Palmer writes. Has for some years now been suggesting a far deeper entanglement between large complex biological organisms and quantum mechanics. Various resear...
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Mathophilia | Math is easy
Algebraic number theory – index. May 5, 2015. This is a list of posts in the algebraic number theory series to date, oldest first. 1 A brief history of algebra. 2 Numbers – rational and irrational, real and imaginary. 4 Groups and rings. 5 Fields and Galois theory. 7 Rings of algebraic integers. 8 Rings and ideals. 9 Uniqueness of factorization. 10 Failure of unique factorization. 11 Why do we care about unique factorization? 12 More concepts from ring theory. 16 Roots of unity and cyclotomic fields.
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