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V is the symmetry group of this cross: flipping it horizontally (a) or vertically (b) or both (ab) leaves it unchanged.A quarter-turn changes it. In two dimensions, the Klein four-group is the symmetry group of a rhombus and of rectangles that are not squares, the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180° rotation.
The minimal polynomial of an element, if it exists, is a member of F[x], the ring of polynomials in the variable x with coefficients in F. Given an element α of E, let J α be the set of all polynomials f(x) in F[x] such that f(α) = 0. The element α is called a root or zero of each polynomial in J α
Chegg, Inc., is an American education technology company based in Santa Clara, California.It provides homework help, digital and physical textbook rentals, textbooks, online tutoring, and other student services.
A field extension L/K is called a simple extension if there exists an element θ in L with L = K ( θ ) . {\displaystyle L=K(\theta ).} This means that every element of L can be expressed as a rational fraction in θ , with coefficients in K ; that is, it is produced from θ and elements of K by the field operations +, −, •, / .
The configurations of the elements in this table are written starting with [Og] because oganesson is expected to be the last prior element with a closed-shell (inert gas) configuration, 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 2 4p 6 4d 10 4f 14 5s 2 5p 6 5d 10 5f 14 6s 2 6p 6 6d 10 7s 2 7p 6. Similarly, the [172] in the configurations for elements ...
Let / be a field extension.An element is a primitive element for / if = (), i.e. if every element of can be written as a rational function in with coefficients in .If there exists such a primitive element, then / is referred to as a simple extension.
An extension of A by B is called split if it is equivalent to the trivial extension. There is a one-to-one correspondence between equivalence classes of extensions of A by B and elements of Ext 1 R (A, B). [9] The trivial extension corresponds to the zero element of Ext 1 R (A, B).
An arbitrary polynomial f with coefficients in some field F is said to have distinct roots or to be square-free if it has deg f roots in some extension field.For instance, the polynomial g(X) = X 2 − 1 has precisely deg g = 2 roots in the complex plane; namely 1 and −1, and hence does have distinct roots.