Polynomial Mod Ideal

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Let us do the above, and further insert some print:

p0, n = 5, 7
L.<x> = PolynomialRing( ZZ )

R = L( x^n - p0^(n-1)*x + p0 )
Rprime = L( n*x^(n-1) - p0^(n-1) )

K.<z> = NumberField( Rprime )
for P in K.primes_of_degree_one_iter():
    print P

The long list ends with:

Fractional ideal (44893, 26936/625*z^5 - 134679/5*z - 8174)
Fractional ideal (49697, -99393/3125*z^5 + 99394/5*z - 24271)
Fractional ideal (71429, -71428/3125*z^5 + 71429/5*z - 13020)

P is now the last entry, it is a fractional ideal. We can factor it like:

sage: P.factor()
(Fractional ideal (7, 7/3125*z^5))^-1 * (Fractional ideal (71429, 7/3125*z^5 - 19711))

Please explain what should be done in this situation. I need one more comment to paste some code...

... continuation:

sage: P
Fractional ideal (71429, -71428/3125*z^5 + 71429/5*z - 13020)
sage: for f, pow in P.factor():
....:     print "pow=%s ideal is %s" % ( pow, f )

pow=-1 ideal is Fractional ideal (7, 7/3125*z^5)
pow=1 ideal is Fractional ideal (71429, 7/3125*z^5 - 19711)

Now f is the last fractional ideal. Its gens are algebraic integers:

sage: for g in f.gens():
....:     print g, 'with norm', ZZ(g.norm()).factor()

71429 with norm 71429^6
7/3125*z^5 - 19711 with norm 2 * 3 * 11311 * 71429 * 12098321599065661

and we may build K.quotient( f ), but this is maybe not wanted.

Passing to ZZ / 71429 first, then modulo ... is what you want?

Also, note that K(R) has degree one, -93750/7*z + 5, we need maybe a new R in K[]...

I'm trying to implement Lemma 2.3 from Kedlaya's A construction of polynomials with squarefree discriminants (I don't believe I can post links, its identifier on Arxiv is 1103.5728). This lemma says "Let $p_0$ be a prime not dividing $n(n−1)$. Then there exist infinitely many primes $p_1$ modulo which the polynomial $R(x) = x^n − p_0^{n−1}x + p_0$ is irreducible and its derivative $R′(x) = nx^{n−1} − p_0^{n−1}$ splits into distinct linear factors.

The proof is:

"The polynomial $R′$ has splitting field $L = \mathbb{Q}(\zeta_n−1, n^{1/(n−1)})$, in which $p_0$ does not ramify because $p_0$ does not divide $n(n−1)$. Thus $R$ is an Eisenstein polynomial with respect to any prime above $p_0$ in $L$; in particular, $R$ is irreducible over $L$. By the Chebotarev density theorem, there exist infinitely many prime ideals of $L$ of absolute degree 1 modulo which $R$ is irreducible; the norm of any such prime ideal is the prime we want."

I'm not entirely sure what "absolute degree 1" means for a prime ideal, and if K.primes_of_degree_one_iter() is giving me something else (namely, fractional ideals), this could be the issue. Do you know how I could find prime ideals of $L$ of absolute degree 1?