[Math] Why is every algebraically closed field perfect

field-theory

I'm having trouble wrapping my head around this, and google has been surprisingly unhelpful. I don't intuitively see why a field being algebraically closed means that every irreducible polynomial over that field must have distinct roots.

Best Answer

Over an algebraically closed field the only irreducible polynomials are those of degree$~1$. They obviously cannot have multiple roots.

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