primitive element of biquadratic field
In other words, is a primitive element (http://planetmath.org/PrimitiveElement) of .
Suppose that . Then . Thus, , which is proven to be false here (http://planetmath.org/QuadraticFieldsThatAreNotIsomorphic). By a , .
Suppose that . Let with , , and such that
Now, we perform some basic algebraic manipulations.
Now, we use equation (1) to eliminate the and obtain
Now, we perform some more basic algebraic manipulations.
Since , , and , we must have . Thus, . (Note that we have since .) Using this in equation (1), we obtain
Now we perform calculations as before.
Since divides and , we must have . Plugging into the equation above yields
Now for yet some more algebraic manipulations.
Thus, or , a contradiction. It follows that . ∎
|Title||primitive element of biquadratic field|
|Date of creation||2013-03-22 17:54:17|
|Last modified on||2013-03-22 17:54:17|
|Last modified by||Wkbj79 (1863)|