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

Latin

Etymology

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

compositum

Noun

compositum n (genitive compositī); second declension

1. that which is agreed; an agreement, compact, etc.
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 compositō (by agreement), as well as more rarely (in the same sense) alone as compositō.

Declension

Second declension.

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[1], 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