Field theory MOC

Field extension

A field extension is the embedding of a field 𝐾 in a larger field 𝐿, #m/def/ring i.e. a Ring monomorphism 𝐾 ↪𝐿 or equivalently 𝐾 is a Subfield of 𝐿. We write 𝐿 :𝐾, and 𝐿 is thence called an extension field of 𝐾. Then 𝐿 may be regarded as a vector space over 𝐾, see Extension field as a unital associative algebra, and its dimension is called the degree of the extension, denoted [𝐿 :𝐾] =dim𝐾⁡𝐿.

A degree 2 extension is called a Quadratic extension, a degree 3 extension is called a Cubic extension, &c.

Types of extension

• Algebraic extension
  • Separable extension
• Simple extension
• Normal extension

See also

• Ring extension
• Adjunction field

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