* Every subextension of F / k is separable.

* Every finite subextension of F / k is separable.

Fix the algebraic closure, and denote by the set of all elements of that are separable over k. is then separable algebraic over k and any separable algebraic subextension of is contaiend in ; it is called the separable closure of k ( inside ).

* The field E < sup > H </ sup > is a normal extension of F ( or, equivalently, Galois extension, since any subextension of a separable extension is separable ) if and only if H is a normal subgroup of Gal ( E / F ).

The one apparent connection between the two is a score of buildings which somehow or other have survived and which naturally enough are called `` historical monuments ''.

The central concern of Erich Auerbach's impressive volume called Mimesis is to describe the shift from a classic theory of imitation ( based upon a recognition of levels of truth ) to a Christian theory of imitation in which the levels are dissolved.

To this end political authority is called upon to exercise its negative and coercive powers.

Even when he is called upon for impromptu remarks, he has notes written on the back of handy envelopes.

Within this frame of reference policies appropriate to claims advanced in the name of the Jews depend upon which Jewish identity is involved, as well as upon the nature of the claim, the characteristics of the claimant, the justifications proposed, and the predispositions of the community decision makers who are called upon to act.

Therefore, what we must prove or disprove is that there were Saxons, in the broad sense in which we must construe the word, in the area of the Saxon Shore at the time it was called the Saxon Shore.

The work as it stands is not the entire book that Malraux wrote at that time -- it is only the first section of a three-part novel called La Lutte avec l'Ange ; ;

And by a skillful and unobtrusive use of imagery ( the enclosure is called a `` Roman-camp stockade '', the hastily erected lean-to is a `` Babylonian hovel '', the men begin to look like `` Peruvian mummies '' and to acquire `` Gothic faces '' ), Malraux projects a fresco of human endurance -- which is also the endurance of the human -- stretching backward into the dark abyss of time.

His very honest act called up the recent talk I had with another minister, a modest Methodist, who said: `` I feel so deeply blessed by God when I can give a message of love and comfort to other men, and I would have it no other way: and it is unworthy to think of self.

RCA Victor has an ambitious and useful project in a stereo series called `` Adventures In Music '', which is an instructional record library for elementary schools.

Where then is the sound planning and cooperation between agencies within the community that you have called for in other editorials??

The `` fruitful course '' of metropolitanization that you recommend is currently practiced by the town of East Greenwich and had its inception long before we learned what it was called.

Here, then, is what Swift would have called a modest proposal by way of a beginning.

The Act further provides for a `` floor '' or minimum allotment, set at the 1954 level, which is called the `` base '' allotment, and a `` ceiling '' or maximum allotment, for each State.

And, given probable public attitudes -- about which reasonably good estimates can be made -- what action is called for to insure necessary support??

Cathy J. Hanover ( Tar Heel-Kaola Hanover ), formerly called Karet Hanover, has been rather a problem child, but is getting better all the while and can pace a twice around in about 2:31.

Ordinary politeness may have militated against this opinion being stated so badly but anyone with a wide acquaintance in both groups and who has sat through the many round tables, workshops or panel discussions -- whatever they are called -- on this subject will recognize that the final, boiled down crux of the matter is education.

This meeting was called to determine how these groups might cooperate to launch what is known as the Outdoor Education Project.

Sixty miles north of New York City where the wooded hills of Dutchess County meet the broad sweep of the Hudson River there is a new home development called `` Oakwood Heights ''.

At the same time, however, I availed myself of the services of that great English actor and master of make-up, Sir Gauntley Pratt, to do a `` quickie '' called The Mystery of the Mad Marquess, in which I played a young American girl who inherits a haunted castle on the English moors which is filled with secret passages and sliding panels and, unbeknownst to anyone, is still occupied by an eccentric maniac.

I called the other afternoon on my old friend, Graves Moreland, the Anglo-American literary critic -- his mother was born in Ohio -- who lives alone in a fairy-tale cottage on the Upson Downs, raising hell and peacocks, the former only when the venerable gentleman becomes an angry old man about the state of literature or something else that is dwindling and diminishing, such as human stature, hope, and humor.

The separable closure is the full algebraic closure if and only if K is a perfect field.

If X ′ is separable, then X is separable.

Then X is separable if and only if X ′ is separable.

* Every compact metric space is separable.

Another active area of research is the program to obtain classification, or to determine the extent of which classification is possible, for separable simple nuclear C *- algebras.

A topological space homeomorphic to a separable complete metric space is called a Polish space.

While most Protestants agree that baptism in the Holy Spirit is integral to being a Christian, others believe that it is not separable from conversion and no longer marked by glossolalia.

** In the product topology, the closure of a product of subsets is equal to the product of the closures.

In fact, it is the smallest algebraically closed field containing the rationals, and is therefore called the algebraic closure of the rationals.

Every field has an algebraic extension which is algebraically closed ( called its algebraic closure ), but proving this in general requires some form of the axiom of choice.

This has changed with the closure of several manufacturers, and Albion's culture is changing to that of a college town with a strong interest in technology and sustainability issues.

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.

Using Zorn's lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.

To see this, note that if L is any algebraic extension of K, then the algebraic closure of L is also an algebraic closure of K, and so L is contained within the algebraic closure of K.

The algebraic closure of K is also the smallest algebraically closed field containing K,

because if M is any algebraically closed field containing K, then the elements of M which are algebraic over K form an algebraic closure of K.

The algebraic closure of a field K has the same cardinality as K if K is infinite, and is countably infinite if K is finite.