When K contains the n-th roots of unity, adjoining an n-th root of a class can produce a cyclic extension. This is a compact note, but the quantifiers and hypotheses stay on the page.
The mathematical object
Kummer equations \(x^n=a\) describe cyclic extensions when roots of unity are available and \(\operatorname{char}K\nmid n\). In characteristic \(p\), Artin--Schreier equations \(x^p-x=a\) play the parallel role.
The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.
One explicit computation
Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.
Why the identity matters
The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.
Where it can fail
Both theories have hypotheses that cannot be removed casually. Missing roots of unity or inseparability changes the Galois group and the classification.
The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.