07-401/Class Notes for March 7
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The information below is preliminary and cannot be trusted! (v)
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[a]/\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.
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].
