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From: JEMebius on 2 Aug 2010 21:50
David Bernier wrote:
> A program tells me that the polynomial p(x) = x^4 + x^3 + x^2 + x + 1 is
> irreducible in Q[x]. If a = exp(2pi i/5) is as usual a 5th root
> of unity, then I think the polynomial p(x) splits or factors in C[x] as:
>
> x^4 + x^3 + x^2 + x + 1 = (x- a)(x - a^2) ( x - a^3) (x - a^4).
>
> I'm trying to understand what algebraists mean by conjugate roots.
>
> So I's like to know which pairs of roots among a, a^2, a^3 and a^4
> are conjugate roots and why.
>
> David Bernier


Key concepts: field extensions; normal field extensions; Galois theory.


An incomplete and somewhat superficial explanation
--------------------------------------------------

One can extend the field Q of rationals to a superfield Q(A) of Q by applying the four
main operations not only the rationals, but by including a single new number

A = exp(2pi.i/5) = [(-1+sqrt(5))/4] + i[sqrt(10+sqrt(20))/4]

in the game.

Q(A) is, like Q, closed under the four main operations: it was constructed that way.
Furthermore, it is the smallest field containing the number A and containing Q as a
subfield. One says: "Q(A) is obtained from Q by adjoining the number A.".
The powers of A are elements of Q(A).

Now for the concept of "conjugate elements":
it happens to be the case that it does not matter which one of the numbers A, A^2, A^3,
A^4 is adjoined to Q; furthermore, these four numbers are all the roots of a single
equation with rational coefficients. Such extensions are named "normal extensions", and
the roots of the defining polynomial (in this case X^4 + X^3 + X^2 + X + 1) belong
together in that sense. Hence the name "conjugate numbers".

Q(A) is transformed into itself by the three mappings
induced by A -> A^2, A -> A^3, A -> A^4.
Together with the identity map (induced by A -> A) they form a group under composition of
mappings, the so-called Galois group of the extension of Q to Q(A), traditionally notated
as (Q(A):Q) or {Q(A):Q} or similar. These transformations are automorphisms of Q(A) which
leave Q element-wise invariant.
Q(A) as a linear space over Q is 4-dimensional. Notation: [Q(A):Q] = 4.
Conjugate elements come in fours.

A more well-known example of conjugacy: x + iy and x - iy (x, y real) in the field of
complex numbers. The field extension is {Q:C}. The base field is R, the superfield is C.
C as an R-linear space is 2-dimensional, in formula: [C:R] = 2. Conjugates come in pairs.

Happy studies: Johan E. Mebius
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