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Old 2015-09-09, 01:51   #8
R.D. Silverman
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Nov 2003

26·113 Posts

Originally Posted by R.D. Silverman View Post
Indeed. This is excellent. Please tell us the method.
One way of doing it for x^2 + 1 is to factor it as (x+i) (x-i), then search for x such that the norms
(over Q) of the factors are both prime.

For an arbitrary quadratic irreducible, factor it over its splitting field, then do as above. It may be hard to
get nearly equal primes, depending on how the poly splits.

For degree k > 2 this becomes more problematic because the polynomial now
splits into k factors.

For degree 4 use Bairstow's method to split it into quadratics [over R, with
algebraic irrational coefficients], then find the pre-image value that renders each quadratic
prime over the splitting field of the quartic. Getting the norms of each factor close may be difficult
if (say) the L2 or L_oo norms of each quadratic are quite different.

Odd degree will be difficult.
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