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How to check the second condition(inertia)?

asked 2022-09-25 15:59:56 +0100

DreDd gravatar image

updated 2022-10-06 16:14:42 +0100

FrédéricC gravatar image

Choose h ∈ Z[t], irreducible of degree η such that p is inert in Q[t]/h(t). How to check the inertia in sage?

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answered 2022-09-25 17:54:45 +0100

rburing gravatar image

updated 2022-09-26 18:01:52 +0100

You want $h$ in $\mathbb{Z}[t]$ such that $h$ is irreducible and the ideal $(p)$ in the ring of algebraic integers of $\mathbb{Q}[t]/h(t)$ is prime. For example, like this:

n = 5
p = 3
R.<t> = PolynomialRing(ZZ)
h = t^n
while not h.is_irreducible() or not NumberField(h, name='m').ideal(p).is_prime():
    h = t^n + R.random_element(degree=n-1)

Example output:

t^5 + t^4 - 5*t^3 + 2*t^2 + t + 2

(In a previous version of this answer I said you want $h \in \mathbb{Z}[t]$ such that both $h$ and $h\text{ mod }p \in \mathbb{F}_p[t]$ are irreducible, but this only works—by the Kummer-Dedekind theorem—if the ring of integers of $K = \mathbb{Q}[t]/h(t)$ is generated by a root of $h$, which is already not the case e.g. for $h = t^2 + 3$ and $p=2$.)

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Thanks, can i ask you one more question? Can we find the roots of h? (Build something like number field Q(m) where m is the root of h)

DreDd gravatar imageDreDd ( 2022-09-25 18:30:48 +0100 )edit

Yes, put K.<a> = h.splitting_field() and then do h.change_ring(K).factor(). Or if you need just one root then factor over L.<m> = NumberField(h).

rburing gravatar imagerburing ( 2022-09-25 23:30:13 +0100 )edit

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Asked: 2022-09-25 15:59:56 +0100

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Last updated: Sep 26 '22