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English
Noun
perfect field (plural perfect fields)
- (algebra, field theory) A field K such that every irreducible polynomial over K has distinct roots.
- 1984, Julio R. Bastida, Field Extensions and Galois Theory, Cambridge University Press, Addison-Wesley, page 10,
- If
is a perfect field of prime characteristic
, and if
is a nonnegative integer, then the mapping
from
to
is an automorphism.
2001, Tsit-Yuen Lam, A First Course in Noncommutative Rings, 2nd edition, Springer, page 116:So far this stronger conjecture has been proved by Nazarova and Roiter over algebraically closed fields, and subsequently by Ringel over perfect fields.
- 2005, Antoine Chambert-Loir, A Field Guide to Algebra, Springer, page 57,
- Definition 3.1.7. One says a field
is perfect if any irreducible polynomial in
has as many distinct roots in an algebraic closure as its degree.
- By the very definition of a perfect field, Theorem 3.1.6 implies that the following properties are equivalent:
- a)
is a perfect field;
- b) any irreducible polynomial of
is separable;
- c) any element of an algebraic closure of
is separable over
;
- d) any algebraic extension of
is separable;
- e) for any finite extension
, the number of
-homomrphisms from
to an algebraically closed extension of
is equal to ![{\displaystyle .</dd></dl></dd>
<dd><b>Corollary 3.1.8.</b> <i>Any algebraic extension of a <b>perfect field</b> is again a <b>perfect field</b>.</i></dd></dl></li></ul></li></ol>
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Usage notes
- A number of simply stated conditions are equivalent to the above definition:
- Every irreducible polynomial over
is separable;
- Every finite extension of
is separable;
- Every algebraic extension of
is separable;
- Either
has characteristic 0, or, if
has characteristic
, every element of
is a
th power;
- Either
has characteristic 0, or, if
has characteristic
, the Frobenius endomorphism
is an automorphism of
;
- The separable closure of
(the unique separable extension that contains all (algebraic) separable extensions of
) is algebraically closed.
- Every reduced commutative K-algebra A is a separable algebra (i.e.,
is reduced for every field extension
).
Hyponyms
Translations
field such that every irreducible polynomial over it has distinct roots
Further reading