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Splitting Fields

Let F be a field and p(x) be a nonconstant polynomial in F[x]. We already know that we can find a field extension of F that contains a root of p(x).

Splitting Fields

Let \(F\) be a field and \(p(x)\) be a nonconstant polynomial in \(F[x]\). We already know that we can find a field extension of \(F\) that contains a root of \(p(x)\). However, we would like to know whether an extension \(E\) of \(F\) containing all of the roots of \(p(x)\) exists. In other words, can we find a field extension of \(F\) such that \(p(x)\) factors into a product of linear polynomials? What is the smallest extension containing all the roots of \(p(x)\)?

Let \(F\) be a field and \(p(x) = a_0 + a_1 x + \cdots + a_n x^n\) be a nonconstant polynomial in \(F[x]\). An extension field \(E\) of \(F\) is a splitting field of \(p(x)\) if there exist elements \(\alpha_1, \ldots, \alpha_n\) in \(E\) such that \(E = F( \alpha_1, \ldots, \alpha_n )\) and \[\begin{aligned}\end{aligned}\]. A polynomial \(p(x) \in F[x]\) splits in \(E\) if it is the product of linear factors in \(E[x]\).

Example

Let \(p(x) = x^4 + 2x^2 - 8\) be in \({\mathbb Q}[x]\). Then \(p(x)\) has irreducible factors \(x^2 -2\) and \(x^2 + 4\). Therefore, the field \({\mathbb Q}( \sqrt{2}, i )\) is a splitting field for \(p(x)\).

Example

Let \(p(x) = x^3 - 3\) be in \({\mathbb Q}[x]\). Then \(p(x)\) has a root in the field \({\mathbb Q}( \sqrt[3]{3}\, )\). However, this field is not a splitting field for \(p(x)\) since the complex cube roots of \(3\), \[\begin{aligned}\end{aligned}\], are not in \({\mathbb Q}( \sqrt[3]{3}\, )\).

Theorem

Let \(p(x) \in F[x]\) be a nonconstant polynomial. Then there exists a splitting field \(E\) for \(p(x)\).

Proof

We will use mathematical induction on the degree of \(p(x)\). If \(\deg p(x) = 1\), then \(p(x)\) is a linear polynomial and \(E = F\). Assume that the theorem is true for all polynomials of degree \(k\) with \(1 \leq k \lt n\) and let \(\deg p(x) = n\). We can assume that \(p(x)\) is irreducible; otherwise, by our induction hypothesis, we are done. By , there exists a field \(K\) such that \(p(x)\) has a zero \(\alpha_1\) in \(K\). Hence, \(p(x) = (x - \alpha_1)q(x)\), where \(q(x) \in K[x]\). Since \(\deg q(x) = n -1\), there exists a splitting field \(E \supset K\) of \(q(x)\) that contains the zeros \(\alpha_2, \ldots, \alpha_n\) of \(p(x)\) by our induction hypothesis. Consequently, \[\begin{aligned}\end{aligned}\] is a splitting field of \(p(x)\).

Condensed: the full section is in Judson, Abstract Algebra: Theory and Applications.

Одоо та Энэ тоог тооцоолох ямар ч калькулятор байхгүй, гэхдээ түүний хэсгүүдийг тооцоолж болно. Доорх нэгийг туршаад үзээрэй эсвэл өөрийнхөө тоог бичнэ үү.

Өөрийнхөө ажлыг үргэлжлүүл

Үнэгүй бүртгэл нь хичээл бүрийн тэмдэглэл, төгссөн хичээл, шийдсэн даалгаврын тэмдэглэл, энэ хуудсан дээр асууж болох багшийг нэгтгэж өгдөг. Математик нь бүртгэлтэй, бүртгэлтэйгүй хэн бүхэнд нээлттэй.

Бүртгүүлэх Нэвтрэх

Энд ашигласан тэмдэгтүүд

Бүх тодорхойлолт, зураг, үсгийн утгыг харахын тулд ямар ч тэмдэгтийг дарна уу.

Хүмүүс асуудаг асуултууд

What is a group, in plain words?

A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group. That is where the idea came from.

What is the difference between a ring and a field?

A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.

Энэ хуудасны зарим хэсгийг Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Энэ нь утгагүй, буруу ойлголт, алдаа юм.

Бүх зүйл Abstract Algebra

Groups Subgroups, cosets and Lagrange's theorem Cyclic groups and permutation groups Homomorphisms, normal subgroups and quotient groups Rings and fields Galois theory: why the quintic has no formula