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The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. [a] A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension. [2]
An extension to the Collatz conjecture is to include all integers, not just positive integers. Leaving aside the cycle 0 → 0 which cannot be entered from outside, there are a total of four known cycles, which all nonzero integers seem to eventually fall into under iteration of f .
Google released a browser extension for the Chrome browser, named with a "beta" tag for unfinished development, shortly thereafter. [120] In May 2014, the company officially added "OK Google" into the browser itself; [ 121 ] they removed it in October 2015, citing low usage, though the microphone icon for activation remained available. [ 122 ]
Extensions of degree 2 and 3 are called quadratic extensions and cubic extensions, respectively. A finite extension is an extension that has a finite degree. Given two extensions L / K {\displaystyle L/K} and M / L {\displaystyle M/L} , the extension M / K {\displaystyle M/K} is finite if and only if both L / K {\displaystyle L/K} and M / L ...
Explosions of electronic devices used by Hezbollah members kill at least 42 people and injure thousands of others in Lebanon and Syria.; Severe flooding (pictured) in central Europe leaves at least 25 people dead and 11 others missing, with several towns submerged.
An early use of (what is now known as) a Kan extension from 1956 was in homological algebra to compute derived functors. In Categories for the Working Mathematician Saunders Mac Lane titled a section "All Concepts Are Kan Extensions", and went on to write that The notion of Kan extensions subsumes all the other fundamental concepts of category ...
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