**
Polynomials of the 3rd Degree:
**

Any **polynomial of the 3rd degree** can be freed from the quadratic term
by a **TSCHIRNHAUS transform**.
(Such transforms have first been studied systematically by
Graf Ehrenfried Walter von TSCHIRNHAUSEN
from Kieslingswalde bey Görlitz.)
Hence we can assume without loss of generality
that the polynomial has the **normal form** P(X) = X^{3} + CX + D with rational integer coefficients C and D.
A **zero** xi of a polynomial of the 3rd degree, which is irreducible over the field Q of rational numbers,
generates a **cubic number field** L = Q(xi) by adjoining xi to Q.
According to the number of real zeros of the generating polynomial P(X), we distinguish
**simply real (or complex)** cubic fields and
**totally real** cubic fields.

**
1. Simply Real Cubic Fields:
**

Over the field R of real numbers, P(X) has a factorization
P(X) = (X - xi) (X^{2} + beta X + gamma)
with the real zero xi in R
and a polynomial of the 2nd degree with coefficients beta,gamma in R which is irreducible over R.
Over the field C of complex numbers, this quadratic factor splits further into
(X - xi') (X - xi'')
with the conjugate complex zeros xi',xi'' in C.
The real cubic number field
L = Q(xi)
has two conjugate isomorphic complex cubic number fields
L' = Q(xi') and L'' = Q(xi'').
The **field monomorphisms** of L consist of
a real inclusion
L-->R, xi-->xi
and a pair of conjugate complex embeddings
L-->C, xi-->xi',
L-->C, xi-->xi''.
The **signature** (s,t) of L is (1,1),
where s denotes the number of real embeddings
and t the number of pairs of conjugate complex embeddings.
Hence, the Unit Theorem of Dirichlet shows
that the unit group U_L of L can be generated
by a single fundamental unit e (of infinite order)
and the root of unity -1:
U_L = < e, -1 >.

**
2. Totally Real Cubic Fields:
**

P(X) splits completely into linear factors over R,
P(X) = (X - xi_{1}) (X - xi_{2})(X - xi_{3})
with the three real zeros xi_{1},xi_{2},xi_{3} in R.
The three zeros generate three conjugate isomorphic real cubic number fields
L_{1} = Q(xi_{1}),
L_{2} = Q(xi_{2}),
L_{3} = Q(xi_{3}).
Among the **field monomorphisms** of L_{1} we have
the real inclusion
L_{1}-->R, xi_{1}-->xi_{1}
and the real embeddings
L_{1}-->R, xi_{1}-->xi_{2},
L_{1}-->R, xi_{1}-->xi_{3}.
Therefore, L_{1} has the **signature**
(s,t) = (3,0).
According to the Unit Theorem of Dirichlet
the unit group U_L of L has
a fundamental system of exactly 2 units e_1 and e_2 (of infinite order)
and the root of unity -1, in this case:
U_L = < e_1, e_2, -1 >.

Like any ABELian group,
the **unit group** U_K of an algebraic number field K splits
into the direct product of the part of finite order and the part of infinite order.
The part of finite order is generally called the **torsion subgroup**
and here specifically the subgroup of the **roots of unity** or torsion units TU_K.
By the **DIRICHLET unit rank r** we understand
the number of free generators of infinite order,
i. e., the Z-Rank of the torsion free part.
The fundamental **Unit Theorem of DIRICHLET** claims
that the DIRICHLET unit rank r can be calculated
from the signature (s,t) in the following manner:
r = s + t - 1.
(The signature is connected with the absolute degree n = [K:Q] of K
by the relation s + 2t = n.)

By a **TSCHIRNHAUS transform** we understand in the most general case the mapping,
which associates the **main polynomial** H_rho(X) of rho
with each element rho of an algebraic number field L:
L --> Q[X], rho --> H_rho(X) = **det( X*1_L - h_rho )**,
where h_rho in End_Q(L) denotes the **homothety** alpha --> rho * alpha.
The TSCHIRNHAUS transform is **independent** from the choice of a **Q-basis** of L.
In particular, it is not affected by
the selection of a **primitive element** delta of L = Q( delta ) with
associated **power basis**
(1,delta,...,delta^{n-1}), where n denotes the degree [L:Q] of L over Q.
For the **actual computation**, however, we choose a primitive element delta of L.
Then we obtain the impression that the TSCHIRNHAUS transform
maps the **minimal polynomial** M_delta(X) of delta onto the main polynomial H_rho(X) of rho:
L * { P(X) in Q[X] | P monic and irreducible of degree n } --> Q[X], ( rho, M_delta(X) ) --> H_rho(X)
where rho = r_{0} + r_{1}*delta + ... + r_{n-1}*delta^{n-1},
since L = Q + Q*delta + ... + Q*delta^{n-1} as a Q-vectorspace,
M_delta(X) = X^{n} + M_{n-1}*X^{n-1} + ... + M_{1}*X + M_{0},
and H_rho(X) = X^{n} + H_{n-1}*X^{n-1} + ... + H_{1}*X + H_{0}.
The coefficients H_{i} of the main polynomial H_rho(X) appear as **polynomials** in the
coordinates r_{i} of rho with respect to delta
and in the coefficients M_{i} of the minimal polynomial M_delta(X).

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