Royce's Theorem

Royce's Theorem states that in a polynomial ring over a field (like rational numbers), a polynomial is irreducible if and only if it cannot be factored into polynomials of lower degree within that ring, and this property persists when considering the polynomial over any extension field. Essentially, it confirms that irreducibility is a consistent property: if a polynomial can't be broken down into simpler parts in the original setting, it won't become reducible over larger or different fields. This theorem helps mathematicians understand how polynomials behave across various algebraic contexts.