Difference between revisions of "Inertial prime number"
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''inert prime number, in an extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508002.png" />'' | ''inert prime number, in an extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508002.png" />'' | ||
− | A prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508003.png" /> such that the principal ideal generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508004.png" /> remains prime in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508005.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508006.png" /> is a finite extension of the field of rational numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508007.png" />; in other words, the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508008.png" /> is prime in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508009.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080010.png" /> is the ring of integers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080011.png" />. In this case one also says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080012.png" /> is inert in the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080013.png" />. By analogy, a prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080014.png" /> of a Dedekind ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080015.png" /> is said to be inert in the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080017.png" /> is the field of fractions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080019.png" /> is a finite extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080020.png" />, if the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080022.png" /> is the integral closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080023.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080024.png" />, is prime. | + | A prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508003.png" /> such that the principal ideal generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508004.png" /> remains prime in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508005.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508006.png" /> is a finite extension of the field of rational numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508007.png" />; in other words, the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508008.png" /> is prime in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i0508009.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080010.png" /> is the ring of integers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080011.png" />. In this case one also says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080012.png" /> is inert in the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080013.png" />. By analogy, a prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080014.png" /> of a Dedekind ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080015.png" /> is said to be inert in the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080017.png" /> is the [[field of fractions]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080019.png" /> is a finite extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080020.png" />, if the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080022.png" /> is the integral closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080023.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080024.png" />, is prime. |
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080025.png" /> is a Galois extension with Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080026.png" />, then for any ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080027.png" /> of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080028.png" />, a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080029.png" /> of the decomposition group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080030.png" /> of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080031.png" /> is defined which is called the inertia group (see [[Ramified prime ideal|Ramified prime ideal]]). The extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080032.png" /> is a maximal intermediate extension in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080033.png" /> in which the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080034.png" /> is inert. | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080025.png" /> is a Galois extension with Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080026.png" />, then for any ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080027.png" /> of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080028.png" />, a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080029.png" /> of the decomposition group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080030.png" /> of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080031.png" /> is defined which is called the inertia group (see [[Ramified prime ideal|Ramified prime ideal]]). The extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080032.png" /> is a maximal intermediate extension in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080033.png" /> in which the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050800/i05080034.png" /> is inert. |
Revision as of 20:57, 28 November 2014
inert prime number, in an extension
A prime number such that the principal ideal generated by
remains prime in
, where
is a finite extension of the field of rational numbers
; in other words, the ideal
is prime in
, where
is the ring of integers of
. In this case one also says that
is inert in the extension
. By analogy, a prime ideal
of a Dedekind ring
is said to be inert in the extension
, where
is the field of fractions of
and
is a finite extension of
, if the ideal
, where
is the integral closure of
in
, is prime.
If is a Galois extension with Galois group
, then for any ideal
of the ring
, a subgroup
of the decomposition group
of the ideal
is defined which is called the inertia group (see Ramified prime ideal). The extension
is a maximal intermediate extension in
in which the ideal
is inert.
In cyclic extensions of algebraic number fields there always exist infinitely many inert prime ideals.
References
[1] | S. Lang, "Algebraic number theory" , Addison-Wesley (1970) |
[2] | H. Weyl, "Algebraic theory of numbers" , Princeton Univ. Press (1959) |
[3] | J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986) |
Comments
Let be a Galois extension with Galois group
. Let
be a prime ideal of (the ring of integers
) of
. The decomposition group of
is defined by
. The subgroup
is the inertia group of
over
. It is a normal subgroup of
. The subfields of
which, according to Galois theory, correspond to
and
, are called respectively the decomposition field and inertia field of
.
Inertial prime number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inertial_prime_number&oldid=35051