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A Salem number of degree **6** has a minimal polynomial of the following form:
where **a,b** and **c** are integers. The trace of is
, where the sum is over the six conjugates
of , i.e., the six roots of (2.1). Two of these conjugates are and
and the remaining conjugates satisfy , so a bound on **|a|** implies
a bound on and hence on **|b|** and **|c|**. So, for fixed **a**, there is a finite
set of with . Observe that
so for all Salem numbers of degree **6**. In fact, there are no
such numbers with trace **-1**, **4** with trace **0**, **15** with trace **1**, and **39** with
trace **2**.

It is not difficult to recognize Salem numbers of degree **6**. Since is
reciprocal, we can write

where
The zeros of **U** are the numbers and hence **U** must have two
roots in the open interval **-2 < x < 2** and one root with **x > 2**. This is equivalent
to the following requirements: (i) , (ii) has real roots,
the smaller of which, satisfies , and . We used
this criterion in compiling our list of Salem numbers.
We also require **U** to be irreducible. Since **U** is cubic, it suffices that for any integer **n**. Only a finite set of **n** need be checked since, by a well-known
estimate of Cauchy, for . It is easy
to see that **P** can only factor into factors of even degree since the roots and
must belong to the same factor, or else there would be a factor the product of
whose roots is in absolute value less than **1**, which is clearly impossible. Thus, **P**
is irreducible if and only if **U** is irreducible.
The following useful result is an elementary deduction from the above discussion.

### Lemma 2.1

Let **P** be as in (2.1) and **U** as in (2.3). A necessary
condition for **P** to be the minimal polynomial of a Salem number is that
for .
A sufficient condition for **P** to be the minimal polynomial of a Salem number is,
in addition to (2.4), any one of , or .

In general, if **P** is the minimal polynomial of a Salem number of degree **d = 2s**, then
there is a monic polynomial **U** with integer coefficients for which . The numbers are the zeros of **U** repeated twice each. The
following computation gives a useful expression for :

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