A biquadratic field (or biquadratic number field) is a biquadratic extension of . To discuss these fields more easily, the set will be defined to be the set of all squarefree integers not equal to . Thus, any biquadratic field is of the form for distinct elements and of .
Let . It can easily be verified that , , and . Since , the three distinct quadratic subfields of are , , and . Note that .
Of the three cyclotomic fields of degree (http://planetmath.org/ExtensionField) four over , and are biquadratic fields. The quadratic subfields of are , , and ; the quadratic subfields of are , , and .
Note that the only rational prime for which is possible in a biquadratic field is . (The notation refers to the ramification index of the prime ideal over .) This occurs for biquadratic fields in which exactly two of , , and are equivalent (http://planetmath.org/Congruences) to and the other is to . For example, in , we have that .
Certain biquadratic fields provide excellent counterexamples to statements that some people might think to be true. For example, the biquadratic field is useful for demonstrating that a subring of a principal ideal domain need not be a principal ideal domain. It can easily be verified that (the ring of integers of ) is a principal ideal domain, but , which is a subring of , is not a principal ideal domain. Also, biquadratic fields of the form with and distinct elements of such that and are useful for demonstrating that rings of integers need not have power bases over (http://planetmath.org/PowerBasisOverMathbbZ). Note that splits completely in both and and thus in . Therefore, for distinct prime ideals (http://planetmath.org/PrimeIdeal) , , , and of . Now suppose for some . Then , and the minimal polynomial for over has degree . This yields that , considered as a polynomial over , is supposed to factor into four distinct monic polynomials of degree , which is a contradiction.
|Date of creation||2013-03-22 15:56:24|
|Last modified on||2013-03-22 15:56:24|
|Last modified by||Wkbj79 (1863)|
|Synonym||biquadratic number field|