82.220.1.204: /* External links */

2013-12-12T13:37:39Z

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			In [[field theory (mathematics)\|field theory]], a branch of algebra, a [[field extension]] <math>L/k</math> is said to be '''regular''' if ''k'' is [[algebraically closed]] in ''L'' {{Clarification needed\|date=July 2013}} and ''L'' is [[separable extension\|separable]] over ''k'', or equivalently, <math>L \otimes_k \overline{k}</math> is an integral domain when <math>\overline{k}</math> is the algebraic closure of <math>k</math> (that is, to say, <math>L, \overline{k}</math> are [[linearly disjoint]] over ''k'').<ref name=FJ38>Fried & Jarden (2008) p.38</ref><ref name=C425>Cohn (2003) p.425</ref>

			==Properties==
			* Regularity is transitive: if ''F''/''E'' and ''E''/''K'' are regular then so is ''F''/''K''.<ref name=FJ39>Fried & Jarden (2008) p.39</ref>
			* If ''F''/''K'' is regular then so is ''E''/''K'' for any ''E'' between ''F'' and ''K''.<ref name=FJ39/>
			* The extension ''L''/''k'' is regular if and only if every subfield of ''L'' finitely generated over ''k'' is regular over ''k''.<ref name=C425/>
			* Any extension of an algebraically closed field is regular.<ref name=FJ39/><ref name=C426>Cohn (2003) p.426</ref>
			* An extension is regular if and only if it is separable and [[primary extension\|primary]].<ref name=FJ44>Fried & Jarden (2008) p.44</ref>
			* A [[purely transcendental extension]] of a field is regular.

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			There is also a similar notion: a field extension <math>L / k</math> is said to be '''self-regular''' if <math>L \otimes_k L</math> is an integral domain. A self-regular extension is relatively algebraically closed in ''k''.<ref name=C427>Cohn (2003) p.427</ref> However, a self-regular extension is not necessarily regular.{{Citation needed\|date=February 2010}}

			==References==
			{{reflist}}
			* {{cite book \| last1=Fried \| first1=Michael D. \| last2=Jarden \| first2=Moshe \| title=Field arithmetic \| edition=3rd revised \| series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge \| volume=11 \| publisher=[[Springer-Verlag]] \| year=2008 \| isbn=978-3-540-77269-9 \| zbl=1145.12001 \| pages=38-41 }}
			* M. Nagata (1985). Commutative field theory: new edition, Shokado. (Japanese) [http://www.shokabo.co.jp/mybooks/ISBN978-4-7853-1309-8.htm]
			* {{cite book \| title=Basic Algebra. Groups, Rings, and Fields \| first=P. M. \| last=Cohn \| authorlink=Paul Cohn \| publisher=[[Springer-Verlag]] \| year=2003 \| isbn=1-85233-587-4 \| zbl=1003.00001 }}
			* A. Weil, [[Foundations of algebraic geometry]].

			[[Category:Field theory]]


			{{Abstract-algebra-stub}}

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2012-08-30T05:26:57Z

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Break the difficulties of conformity as well as go all from a biking journey. If you have any inquiries pertaining to where by and how to use [http://www.wallpaperhdquality.com/profile/toliddell Cannondale mountain bike sizing.], you can call us at our own web-site. You'll ask yourself this question in conjunction with "how much will you pay. High end mountain bikes use carbon fibre frames, or other, more exotic materials to reduce weight and keep stiffness up. The remarkable thing though was an electric bike could cover 25 miles on one charge with a bet of pedal power or up to 15 miles on battery power lone. Look at what other riders appreciate about their bikes. Adjust the tyre pressure too so it is at its optimum level. Once you're comfortable coasting, dropping, standing, pedaling, spinning, and switching gears, you'll be ready to hit the trails, and tackle any challenge along the way. Each rider has lanes, but some others only have a single lane. If biking is your passion or your favorite sport, then you know the worth of good accessories while riding. If you are riding through a rock garden, for instance, you don't want a fork that is bouncing you all over the place. Some also use the liner as a size aid, for example BMX helmets can come as one-size, with optional liner kits to change the helmets size. An Indian travois style trailer is very simple to make and very effective for heavy loads. But when it comes to mountain biking it is not enough that you own a bike. Another way to classify brakes is by mounting style. These are stable forks whose weights are directly in proportion to their durability. They are usually slightly used and cheaper than a brand new bike. Bikes are a great way to get around - they're fun, they're cheap to fuel (they burn only calories) and they're a great way to get fit. Maybe you like to go out; maybe you like to mountain bike. Putting is difficult, when you are just starting out getting on the green is great, but as you get better you want to put your ball as close to the pin as you can. In addition, being positioned over the rear tire makes for an extremely light front end, which is hard to hold down on inclines. No matter what type of bike, no matter how old it is, no matter how much money you'd like to spend. Try finding the best bike for yourself at local shops, where you can take a test ride before you finally decide. The best way of getting mountain bike components is shopping online. It is also easier to hold the handlebars if they are covered with rubber than exposed handlebars, especially if it is too hot because steel absorbs heat and makes it difficult for you to hold. As with all kinds of products, the prices for kids dirt bike usually vary depending on the size, style, materials and of course manufacturer.

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