Re: a question about algebraic numbers

On Sun, 02 Apr 2006 18:28:17 -0400, quasi <quasi@xxxxxxxx> wrote:

On Sun, 02 Apr 2006 18:00:34 EDT, Feng <lux@xxxxxxxxxxxx> wrote:

From others's post, I am considering a question, if a and b in C (complex numbers)are both algebraic, then what can we know about ab?
ab is also algebraic, that is obvious, anything else?
for example, is there any bound of the degree?

deg(ab) <= deg(a)*deg(b)

The degree of the product is bounded above by the product of the
degrees of the factors.

The proof is immediate -- just consider the chain of fields Q < Q(a) <
Q(a,b).

Less immediate, but true, is the following:

Theorem:

Let a,b be algebraic, and let m=deg(a), n=deg(b).

If (m,n)=1 then deg(a+b)=m*n.

I don't recall the proof, but I remember that it used some Galois
theory.

Is the same result true for the product?

Question:

Let a,b be algebraic, and let m=deg(a), n=deg(b).

If (m,n)=1 must deg(a*b)=m*n?

quasi
.

Relevant Pages

  • RECALL: Toys "R" Us - Wooden Coloring Cases
    ... NEWS from CPSC ... U.S. Consumer Product Safety Commission ... FOR IMMEDIATE RELEASE ... Firm's Recall Hotline: 869-7787 ...
    (misc.consumers)
  • RECALL: Jakes Fireworks (fireworks are fire hazard???)
    ... U.S. Consumer Product Safety Commission ... FOR IMMEDIATE RELEASE ... Firm's Recall Hotline: 766-1277 ... Jakes Fireworks Inc. Recalls Fireworks Sold by World Class ...
    (misc.consumers)
  • Re: September 11
    ... Base went to immediate ... wartime alert with full recall. ... Japanese ships as well with us (combined ...
    (alt.home.repair)
  • Re: HSM Recalls
    ... If the files are not very large I would recommend keeping them on disk ... That would solve the immediate issue. ... that needs recalling following the place where RECALL is invoked. ... message is not the intended recipient or an authorized representative of ...
    (bit.listserv.ibm-main)
  • Re: Functions and Curves
    ... quasi wrote: ... you recall, when I made the initial claim, it _was_ challenged, which ... I was reserving my judgment ... an argument that convinces me either way." ...
    (sci.math)
Quantcast