compositum

English

Etymology

From Latin compositum.

Noun

compositum (plural compositums)

1. (algebra, field theory) Given a field extension F/K and subextensions A and B, the smallest subextension that contains both A and B.
   • 1998, Iain T. Adamson (translator), David Hilbert, The Theory of Algebraic Number Fields, [1897, D. Hilbert, Zahlbericht], Springer, page 98,

     Of particular interest is the case in which the discriminants of the fields forming the compositum are relatively prime.

   • 2004, Dinesh S. Thakur, Function Field Arithmetic, World Scientific, page 81:

     For ${\displaystyle \mathbb {Q} }$, we get the maximal abelian extension by adjoining all roots of unity, i.e., taking compositum of all ${\displaystyle m}$-th cyclotomic fields (this is the famous Kronecker-Weber theorem).

   • 2005, T. Y. Lam, Introduction to Quadratic Forms over Fields, American Mathematical Society, page 333:

     This is then just the field compositum of all the quadratic extensions ${\displaystyle F({\sqrt {a_{i}}})}$ in the algebraic closure of ${\displaystyle F}$.

Usage notes

• Denoted ${\displaystyle A\cdot B}$.
• It is the same as the image of the homomorphism ${\displaystyle h:(A\otimes _{k}B)\to F}$ that maps the tensor product ${\displaystyle a\otimes b\to ab}$.

Synonyms

• (smallest subextension of a given field extension that contains two given subextensions): field compositum

Translations

smallest subextension of a given field extension that contains two given subextensions

Further reading

• Tensor product of fields on Wikipedia.Wikipedia
• Compositum on Encyclopedia of Mathematics

Latin

Etymology

Inflected form of compositus; the noun is the substantivised neuter form.

Adjective

compositum

1. inflection of compositus:
   1. nominative/accusative/vocative neuter singular
   2. accusative masculine singular

Noun

compositum n (genitive compositī); second declension

1. that which is agreed; an agreement, compact, etc.

   Synonyms: condiciō, stipulātiō, pactum

2. compound word, compound

Usage notes

• This noun also appears (in the ablative only) in the phrases ēx compositō (“according to agreement”, “by agreement”, “in concert”) and dē compositō (“by agreement”), as well as more rarely (in the same sense) alone as compositō.

Declension

Second-declension noun (neuter).

Case	Singular	Plural
Nominative	compositum	composita
Genitive	compositī	compositōrum
Dative	compositō	compositīs
Accusative	compositum	composita
Ablative	compositō	compositīs
Vocative	compositum	composita

References

• “compositum”, in Charlton T. Lewis (1891) An Elementary Latin Dictionary, New York: Harper & Brothers
• compositum in Gaffiot, Félix (1934) Dictionnaire illustré latin-français, Hachette.
• Carl Meißner, Henry William Auden (1894) Latin Phrase-Book‎, London: Macmillan and Co.
  • (ambiguous) well-ordered, well-brushed hair: capilli compti, compositi (opp. horridi)
    
  • (ambiguous) an elaborate speech: oratio composita
    
  • (ambiguous) well-arranged words: verba composita