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A polynomial of degree two or three over a field is irreducible exactly when it has no root in the field
Statement
Let be a field and let have degree or . Then is irreducible over if and only if has no root in .
Facts & Assumptions
Given: A field and a polynomial of degree or .
An element is a root of exactly when divides (Factor theorem over a commutative ring).
Degrees add in a product of nonzero polynomials over a field (Over an integral domain, degrees add under multiplication of nonzero polynomials).
The units of are exactly the nonzero constants (The units of over an integral domain are exactly the constant polynomials whose values are units of ).
A nonzero nonunit is irreducible exactly when every factorization has a unit factor (Irreducible and prime elements of an integral domain).
Proof
If has a root , then [L1] gives ; [L2] and make both factors nonunits by [L3], so [L4] shows that is reducible.
Conversely, if with both factors nonunits, [L2] and [L3] give positive degrees summing to or , so one factor has degree ; writing it as with , it has root , and that root is a root of . Thus reducibility implies a root, proving the biconditional.
Depends on
Used by
- A normal basis of F₈ over F₂ Example
- Gal(F₈/F₂) is cyclic of order three with no proper intermediate field Example
- Over F₂, x⁴+x²+1=(x²+x+1)² has two distinct roots, each repeated, in its four-element splitting field Example
- Reduction modulo 2 proves x³+17x+391 irreducible over ℚ Example
- The divisor-sum identity at q=2, n=3 finds exactly two monic irreducible cubics Example
- The four roots of t⁴+t+1 over F₂ are the Frobenius powers of any one of them Example
- The four-element field (ℤ/2)[x]/(x²+x+1) Example
- The polynomial x²-2 is irreducible over ℚ Example
- The polynomial x²+1 is irreducible over ℝ Example
- FALSE: every basis of a finite field over a subfield is a normal basis False statement
- x²+1 is irreducible over ℝ Lemma
Dependency tree · two levels
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Sources
- Neil Donaldson, Math 120B Notes, Theorem 23.8 (standard reference, not scraped)