Search results
Results from the WOW.Com Content Network
The coefficient of the highest-degree term in the polynomial is required to be 1. More formally, a minimal polynomial is defined relative to a field extension E/F and an element of the extension field E/F. 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.
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 gaps in his sketch could easily be filled [3] (as remarked by the referee Poisson) by exploiting a theorem [4] [5] of Lagrange from 1771, which Galois certainly knew. It is likely that Lagrange had already been aware of the primitive element theorem for splitting fields. [5] Galois then used this theorem heavily in his development of the ...
A Kummer extension is a field extension L/K, where for some given integer n > 1 we have K contains n distinct nth roots of unity (i.e., roots of X n − 1) L/K has abelian Galois group of exponent n. For example, when n = 2, the first condition is always true if K has characteristic ≠ 2.
If E is an extension of F in which f is a product of linear factors then no square of these factors divides f in E[X] (that is f is square-free over E). [6] There exists an extension E of F such that f has deg(f) pairwise distinct roots in E. [6] The constant 1 is a polynomial greatest common divisor of f and f '. [7]
An algebraic extension is a purely inseparable extension if and only if for every , the minimal polynomial of over F is not a separable polynomial. [1] If F is any field, the trivial extension is purely inseparable; for the field F to possess a non-trivial purely inseparable extension, it must be imperfect as outlined in the above section.
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).
The order of an element of a group (also called period length or period) is the order of the subgroup generated by the element. If the group operation is denoted as a multiplication , the order of an element a of a group, is thus the smallest positive integer m such that a m = e , where e denotes the identity element of the group, and a m ...