07-401/Class Notes for March 7

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Class Plan

Some discussion of the term test and HW6.

Some discussion of our general plan.

Extension Fields

Definition. An extension field [math]\displaystyle{ E }[/math] of [math]\displaystyle{ F }[/math].

Theorem. For every non-constant polynomial [math]\displaystyle{ f }[/math] in [math]\displaystyle{ F[x] }[/math] there is an extension [math]\displaystyle{ E }[/math] of [math]\displaystyle{ F }[/math] in which [math]\displaystyle{ f }[/math] has a zero.

Example [math]\displaystyle{ x^2+1 }[/math] over [math]\displaystyle{ {\mathbb R} }[/math].

Example [math]\displaystyle{ x^5+2x^2+2x+2=(x^2+1)(x^3+2x+2) }[/math] over [math]\displaystyle{ {\mathbb Z}/3 }[/math].

Definition. [math]\displaystyle{ F(a_1,\ldots,a_n) }[/math].

Theorem. If [math]\displaystyle{ a }[/math] is a root of an irreducible polynomial [math]\displaystyle{ p\in F[x] }[/math], within some extension field [math]\displaystyle{ E }[/math] of [math]\displaystyle{ F }[/math], then [math]\displaystyle{ F(a)\cong F[x]/\langle p\rangle }[/math], and [math]\displaystyle{ \{1,a,a^2,\ldots,a^{n-1}\} }[/math] (here [math]\displaystyle{ n=\deg p }[/math]) is a basis for [math]\displaystyle{ F(a) }[/math] over [math]\displaystyle{ F }[/math].

Corollary. In this case, [math]\displaystyle{ F(a) }[/math] depends only on [math]\displaystyle{ p }[/math].

Splitting Fields

Definition. [math]\displaystyle{ f\in F[x] }[/math] splits in [math]\displaystyle{ E/F }[/math], a splitting field for [math]\displaystyle{ f }[/math] over [math]\displaystyle{ F }[/math].

Theorem. A splitting field always exists.

Example. [math]\displaystyle{ x^4-x^2-2=(x^2-2)(x^2+1) }[/math] over [math]\displaystyle{ {\mathbb Q} }[/math].

Example. Factor [math]\displaystyle{ x^2+x+2\in{\mathbb Z}_3[x] }[/math] within its splitting field [math]\displaystyle{ {\mathbb Z}_3[x]/\langle x^2+x+2\rangle }[/math].

Theorem. Any two splitting fields for [math]\displaystyle{ f\in F[x] }[/math] over [math]\displaystyle{ F }[/math] are isomorphic.

Lemma 1. If [math]\displaystyle{ p\in F[x] }[/math] irreducible over [math]\displaystyle{ F }[/math], [math]\displaystyle{ \phi:F\to F' }[/math] an isomorphism, [math]\displaystyle{ a }[/math] a root of [math]\displaystyle{ p }[/math] (in some [math]\displaystyle{ E/F }[/math]), [math]\displaystyle{ a' }[/math] a root of [math]\displaystyle{ \phi(p) }[/math] in some [math]\displaystyle{ E'/F' }[/math], then [math]\displaystyle{ F(a)\cong F'(a') }[/math].

Lemma 2. Isomorphisms can be extended to splitting fields.

Zeros of Irreducible Polynomials

Definition. The derivative of a polynomial.

Claim. The derivative operation is linear and satisfies Leibnitz's law.

Theorem. [math]\displaystyle{ f\in F[x] }[/math] has a multiple zero in some extension field of [math]\displaystyle{ F }[/math] iff [math]\displaystyle{ f }[/math] and [math]\displaystyle{ f' }[/math] have a common factor of positive degree.

Lemma. The property of "being relatively prime" is preserved under extensions.

Theorem. Let [math]\displaystyle{ f\in F[x] }[/math] be irreducible. If [math]\displaystyle{ \operatorname{char}F=0 }[/math], then [math]\displaystyle{ f }[/math] has no multiple zeros in any extension of [math]\displaystyle{ F }[/math]. If [math]\displaystyle{ \operatorname{char}F=p\gt 0 }[/math], then [math]\displaystyle{ f }[/math] has multiple zeros (in some extension) iff it is of the form [math]\displaystyle{ g(x^p) }[/math] for some [math]\displaystyle{ g\in F[x] }[/math].

Definition. A perfect field.

Theorem. A finite field is perfect.

Theorem. An irreducible polynomial over a perfect field has no multiple zeros (in any extension).

Theorem. Let [math]\displaystyle{ f\in F[x] }[/math] be irreducible and let [math]\displaystyle{ E }[/math] be the splitting field of [math]\displaystyle{ f }[/math] over [math]\displaystyle{ F }[/math]. Then in [math]\displaystyle{ E }[/math] all zeros of [math]\displaystyle{ f }[/math] have the same multiplicity.

Corollary. [math]\displaystyle{ f }[/math] as above must have the form [math]\displaystyle{ a(x-a_1)^n\cdots(x-a_k)^n }[/math] for some [math]\displaystyle{ a\in F }[/math] and [math]\displaystyle{ a_1,\ldots,a_k\in E }[/math].

Example. [math]\displaystyle{ x^2-t\in{\mathbb Z}_2(t)[x] }[/math] is irreducible and has a single zero of multiplicity 2 within its splitting field over [math]\displaystyle{ {\mathbb Z}_2(t)[x] }[/math].