proof of prime ideal decomposition in quadratic extensions of
Much of the proof of this theorem is given in Marcus’ Number Fields (http://planetmath.org/NumberField); however, all of the details will be filled in here, and some aspects of the proof here will differ from those of Marcus.
If is a rational prime that divides , then
Note that . (If they were equal, then would equal .)
If , then . Note that divides . Thus, ramifies in . Therefore, for some prime ideal of . Moreover, is the unique ideal of of norm (http://planetmath.org/IdealNorm) . Since , then
Since has , it follows that and .
If , then . Note that does not divide . Thus, does not ramify in . Since
we have that and must be distinct. Proving that these ideals are indeed given below.
If , then consider the minimal polynomial for . Since , it must be the case that .
Let be a lying over in . Note that has a root (http://planetmath.org/Root) in and thus in . On the other hand, since , considered as an element of has no root in . Thus, and are not isomorphic. Therefore, . Since , we have that . Thus, is inert in . It follows that is in .
If is an odd prime (http://planetmath.org/Prime) that does not divide and , then does not divide (which equals either or ). Thus, does not ramify in . Also, does not divide . Since
we have that and must be distinct. It will be proven that these ideals are indeed .
Let denote the norm of the ideal (http://planetmath.org/IdealNorm) of and with . Then
Note that . Therefore, . It follows that the indicated ideals are .
Finally, if is an odd prime that does not divide and is not a square , then consider the minimal polynomial for over . Let be a lying over in . Note that has a root in and thus in . On the other hand, since , which is not a square in , then considered as an element of has no root in . Thus, and are not isomorphic. Therefore, . Note that . Thus, . Therefore, is inert in . It follows that is in . ∎
- 1 Marcus, Daniel A. Number Fields. New York: Springer-Verlag, 1977.
|Title||proof of prime ideal decomposition in quadratic extensions of|
|Date of creation||2013-03-22 15:59:06|
|Last modified on||2013-03-22 15:59:06|
|Last modified by||Wkbj79 (1863)|