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Lettris is a curious tetris-clone game where all the bricks have the same square shape but different content. Each square carries a letter. To make squares disappear and save space for other squares you have to assemble English words (left, right, up, down) from the falling squares.
Boggle gives you 3 minutes to find as many words (3 letters or more) as you can in a grid of 16 letters. You can also try the grid of 16 letters. Letters must be adjacent and longer words score better. See if you can get into the grid Hall of Fame !
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Otherwise, k is called imperfect.
In particular, all fields of characteristic zero and all finite fields are perfect.
Perfect fields are significant because Galois theory over these fields becomes simpler, since the general Galois assumption of field extensions being separable is automatically satisfied over these fields (see third condition above).
More generally, a ring of characteristic p (p a prime) is called perfect if the Frobenius endomorphism is an automorphism. (This is equivalent to the above condition "every element of k is a pth power" for integral domains.)
Examples of perfect fields are:
In fact, most fields that appear in practice are perfect. The imperfect case arises mainly in algebraic geometry in characteristic p>0. Every imperfect field is necessarily transcendental over its prime subfield (the minimal subfield), because the latter is perfect. An example of an imperfect field is
The first condition says that, in characteristic p, a field adjoined with all pr-th roots (r≥1) is perfect; it is called the perfect closure of k and usually denoted by .
In terms of universal properties, the perfect closure of a ring A of characteristic p is a perfect ring Ap of characteristic p together with a ring homomorphism u : A → Ap such that for any other perfect ring B of characteristic p with a homomorphism v : A → B there is a unique homomorphism f : Ap → B such that v factors through u (i.e. v = fu). The perfect closure always exists; the proof involves "adjoining p-th roots of elements of A", similar to the case of fields.
The perfection of a ring A of characteristic p is the dual notion (though this term is sometimes used for the perfect closure). In other words, the perfection R(A) of A is a perfect ring of characteristic p together with a map θ : R(A) → A such that for any perfect ring B of characteristic p equipped with a map φ : B → A, there is a unique map f : B → R(A) such that φ factors through θ (i.e. φ = θf). The perfection of A may be constructed as follows. Consider the projective system
where the transition maps are the Frobenius endomorphism. The inverse limit of this system is R(A) and consists of sequences (x0, x1, ... ) of elements of A such that for all i. The map θ : R(A) → A sends (xi) to x0.