A quotient that does not preserve lower ramification groups #
Embed ℚ₂(√2) in ℚ₂(ζ₈) by √2 ↦ ζ₈ - ζ₈³ = ζ₈ + ζ₈⁻¹.
The image is the fixed field of the involution ζ₈ ↦ ζ₈⁻¹.
Restriction maps the third lower ramification group of the cyclotomic extension onto the
whole quadratic Galois group, whereas the third lower group of the quadratic extension is
trivial. This gives a concrete failure of compatibility of lower numbering with quotients.
The restriction map is Mathlib's AlgEquiv.restrictNormalHom for this embedding. The lower
filtrations are the canonical ones computed in the two imported worked examples.
References #
- J.-P. Serre, Local Fields, Chapter IV, §§1 and 3.
The embedding ℚ₂(√2) → ℚ₂(ζ₈) sending √2 to ζ₈ - ζ₈³.
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The image of the distinguished quadratic generator.
Restriction from the cyclotomic Galois group to its √2 subextension.
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Restriction commutes with the chosen embedding.
Every quadratic automorphism extends to the cyclotomic field.
The exponent-seven involution, sending ζ₈ to ζ₈⁻¹.
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The exponent-seven automorphism inverts the distinguished root.
The exponent-seven automorphism has order two.
The embedded quadratic field is generated by ζ₈ - ζ₈³.
The fixed field of the exponent-seven involution is the embedded copy of ℚ₂(√2).
The kernel of quadratic restriction is generated by the exponent-seven involution.
The inversion subgroup is normal, as the kernel of quadratic restriction.
Restriction identifies the quotient by inversion with the quadratic Galois group.
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On a cyclotomic automorphism's class, the quotient equivalence is restriction.
The restriction of the exponent-five automorphism negates √2.
The third cyclotomic lower ramification group restricts onto the whole quadratic group.
Lower numbering fails to commute with this restriction: the image at depth three is the whole quadratic Galois group, but the quadratic lower group at depth three is trivial.
In quotient notation, G₃H/H = G/H for the inversion subgroup H.