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In this case, L is an extension field of K and K is a subfield of L. [1] [2] [3] For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.
In mathematical logic, a theory can be extended with new constants or function names under certain conditions with assurance that the extension will introduce no contradiction. Extension by definitions is perhaps the best-known approach, but it requires unique existence of an object with the desired property. Addition of new names can also be ...
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 +, −, •, / .
Alternatively, the Yoneda product can be defined in terms of resolutions. (This is close to the definition of the derived category.) For example, let R be a ring, with R-modules A, B, C, and let P, Q, and T be projective resolutions of A, B, C. Then Ext i R (A,B) can be identified with the group of chain homotopy classes of chain maps P → Q[i ...
An algebraically closed field F has no proper algebraic extensions, that is, no algebraic extensions E with F < E. [10] An example is the field of complex numbers. 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. [11]
Let L be an extension of a field K. Then: If L is a normal extension of K and if E is an intermediate extension (that is, L ⊇ E ⊇ K), then L is a normal extension of E. [4] If E and F are normal extensions of K contained in L, then the compositum EF and E ∩ F are also normal extensions of K. [4]
In mathematical logic, more specifically in the proof theory of first-order theories, extensions by definitions formalize the introduction of new symbols by means of a definition. For example, it is common in naive set theory to introduce a symbol ∅ {\displaystyle \emptyset } for the set that has no member.
The extensional definition of function equality, discussed above, is commonly used in mathematics. A similar extensional definition is usually employed for relations: two relations are said to be equal if they have the same extensions.