Galois saw that every equation hides a group of symmetries and that group alone decides whether the roots can ever be written with radicals.
The field ℚ(√2, i) and its three quadratic subfields correspond exactly to the subgroups of the Klein four-group. Blue labels mark the fixed fields; the lattice is the dictionary between algebra and geometry.
For the quintic x⁵ − 4x + 2, Eisenstein shows it is irreducible and it has precisely three real roots. Complex conjugation therefore acts as a transposition, forcing the Galois group to be all of S₅. Since A₅ is simple, S₅ is not solvable, so the roots lie in no radical tower.
Solvability is not about clever formulas. It is a structural property of the symmetry group.
The wounded are being evacuated from Dnipro railway station to hospitals.
My account on X appears to be throttled, so please share this. People need to see this blatant attack on civilians.
I’m heading down there shortly.
Unbelievably sickening. I’m writing an article right now, having already warned about the dangerous escalation and direct targeting of railway stations and passenger trains.
Explosions are still continuing now….
Now, 13 people have been killed in my city, civilians 🤬
@VillagePrisoner@Historionome@GanzeGeschichte Vielleicht bin ich da auch weniger sprachbegabt, aber ich verstehe das Platt in meinem hessischen Heimatdorf, dass ich im Gegensatz zu meinen älteren Geschwistern nicht gelernt habe, nicht 🤷🏻♂️
Why can one measure be completely recovered from another using only a single density function?
The Radon–Nikodym theorem guarantees that whenever a measure is absolutely continuous with respect to a positive σ-finite measure, it can be written as the integral of an integrable function (the density) against the base measure. Absolute continuity simply means the second measure never places mass where the first has none.
This single idea underlies importance sampling in machine learning, change-of-measure methods in quantitative finance, and likelihood ratios throughout statistics and signal processing.
Mathematics turns the qualitative notion of “more concentrated here” into a concrete function that algorithms can compute and reweight.