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Suppose that E/F is a field extension. Then E may be considered as a vector space over F (the field of scalars). The dimension of this vector space is called the degree of the field extension, and it is denoted by [E:F]. The degree may be finite or infinite, the field being called a finite extension or infinite extension accordingly.
Another extension field of the rationals, which is also important in number theory, although not a finite extension, is the field of p-adic numbers for a prime number p. It is common to construct an extension field of a given field K as a quotient ring of the polynomial ring K [ X ] in order to "create" a root for a given polynomial f ( X ).
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 +, −, •, / .
Thus, a field extension is a transcendental extension if and only if its transcendence degree is nonzero. Transcendental extensions are widely used in algebraic geometry . For example, the dimension of an algebraic variety is the transcendence degree of its function field .
An algebraic number field (or simply number field) is a finite-degree field extension of the field of rational numbers. Here degree means the dimension of the field as a vector space over Q {\displaystyle \mathbb {Q} } .
Extension field If F is a subfield of E then E is an extension field of F. We then also say that E/F is a field extension. Degree of an extension Given an extension E/F, the field E can be considered as a vector space over the field F, and the dimension of this vector space is the degree of the extension, denoted by [E : F]. Finite extension
The notion of a subfield E ⊂ F can also be regarded from the opposite point of view, by referring to F being a field extension (or just extension) of E, denoted by F / E, and read "F over E". A basic datum of a field extension is its degree [F : E], i.e., the dimension of F as an E-vector space. It satisfies the formula [30]
The field L is a normal extension if and only if any of the equivalent conditions below hold. The minimal polynomial over K of every element in L splits in L ; There is a set S ⊆ K [ x ] {\displaystyle S\subseteq K[x]} of polynomials that each splits over L , such that if K ⊆ F ⊊ L {\displaystyle K\subseteq F\subsetneq L} are fields, then ...