primitive element of biquadratic field


Let m and n be distinct squarefreeMathworldPlanetmath integers, neither of which is equal to 1. Then the biquadratic field Q(m,n) is equal to Q(m+n).

In other words, m+n is a primitive elementMathworldPlanetmathPlanetmath ( of (m,n).


We clearly have (m+n)(m,n). For the reverse inclusion, it is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath ( to show that m+n does not belong to any of the quadratic subfieldsMathworldPlanetmath of (m,n), which are (m), (n), and (mn).

Suppose that m+n(m). Then n(m). Thus, (n)=(m), which is proven to be false here ( By a , m+n(n).

Suppose that m+n(mn). Let a,b,c,d with gcd(a,b)=gcd(c,d)=1, b0, and d0 such that

m+n=ab+cdmn. (1)

Now, we perform some basic algebraic manipulations.

bdm+bdn =ad+bcmn
bdm+bdn-ad =bcmn
(bdm+bdn-ad)2 =(bcmn)2
b2d2m+2b2d2mn-2abd2m+b2d2n-2abd2n+a2d2 =b2c2mn
b2d2m+b2d2n+a2d2-b2c2mn =2abd2(m+n)-2b2d2mn

Now, we use equation (1) to eliminate the m+n and obtain

b2d2m+b2d2n+a2d2-b2c2mn =2abd2(ab+cdmn)-2b2d2mn.

Now, we perform some more basic algebraic manipulations.

b2d2m+b2d2n+a2d2-b2c2mn =2a2d2+2abcdmn-2b2d2mn
b2d2m+b2d2n-a2d2-b2c2mn =2bd(ac-bd)mn

Since mn, b0, and d0, we must have ac-bd=0. Thus, cd=ba. (Note that we have a0 since ac=bd0.) Using this in equation (1), we obtain


Now we perform calculations as before.

abm+abn =a2+b2mn
abm+abn-a2 =b2mn
(abm+abn-a2)2 =(b2mn)2
a2b2m+2a2b2mn-2a3bm+a2b2n-2a3bn+a4 =b4mn
a2b2m+a2b2n+a4-b4mn =2a3b(m+n)-2a2b2mn
a2b2m+a2b2n+a4-b4mn =2a3b(ab+bamn)-2a2b2mn
a2b2m+a2b2n+a4-b4mn =2a4+2a2b2mn-2a2b2mn
a2b2m+a2b2n-a4-b4mn =0

Since b2 divides a4 and gcd(a,b)=1, we must have b2=1. Plugging into the equation above yields


Now for yet some more algebraic manipulations.


Thus, m=a2 or n=a2, a contradictionMathworldPlanetmathPlanetmath. It follows that (m+n)=(m,n). ∎

Title primitive element of biquadratic field
Canonical name PrimitiveElementOfBiquadraticField
Date of creation 2013-03-22 17:54:17
Last modified on 2013-03-22 17:54:17
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 9
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 11R16
Related topic PrimitiveElementTheorem