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Chapter 7: Problem 11

Let \(K / F\) be a Galois extension whose Galois group is the symmetric group on 3 elements. Prove that \(K\) does not contain a cyclic extension of \(F\) of degree 3. How many non-cyclic cxtensions of degree 3 does \(K\) contain?

Short Answer

Expert verified
\(K\) does not contain a cyclic extension of \(F\) of degree 3 because none of the cyclic subgroups of the Galois group \(S_3\) correspond to a cyclic extension of degree 3. There is exactly 1 non-cyclic extension of \(F\) of degree 3 contained in \(K\), which comes from the non-cyclic subgroup \(A_3\) (the alternating group of order 3).

Step by step solution

01

Analyze the Galois group S_3

The symmetric group on 3 elements, denoted as \(S_3\), has the following properties: - Order: \(|S_3| = 3! = 6\) - Subgroups: \(S_3\) has only 2 normal subgroups: the trivial group \(\{e\}\) (identity) and the alternating group \(A_3\) of order 3. There are 4 more subgroups, all cyclic of order 2. Now we will use this information to study the intermediate fields of the Galois extension \(K/F\).
02

Analyze the intermediate fields of K/F

For a Galois extension \(K/F\) with Galois group \(G\), there is a one-to-one correspondence between intermediate fields \(E\) with \(F \subseteq E \subseteq K\) and subgroups of \(G\) as per the fundamental theorem of Galois theory. Since \(G=S_3\) has 6 subgroups, there are also 6 intermediate fields of \(K/F\). Note that among those six subgroups, there are four of order 2 (cyclic). It's important to note that whenever the Galois group of an extension \(L/F\) is cyclic, then the extension is a cyclic extension. Let's check if any of these cyclic subgroups of order 2 gives a cyclic extension of degree 3.
03

Show K does not contain a cyclic extension of F of degree 3

Recall that for any subgroup \(H\) of \(G(S_3)\), there exists a corresponding intermediate field \(E\) such that \([K:E]=|H|\) and \([E:F]=\frac{|G|}{|H|}\). Now consider the cyclic subgroups of order 2. As the order of each of these subgroups is 2, we have: \[ [K:E]=|H|=2 \] and \[ [E:F]=\frac{|G|}{|H|}=\frac{6}{2}=3 \] The degrees of the extensions follow the formula \([K:F]=[K:E][E:F]\), which means that for each cyclic subgroup H of order 2, \[ [K:F]=[K:E][E:F]=2 \cdot 3=6 \] However, we are looking for a cyclic extension of degree 3. As we can see, none of the four cyclic subgroups (order 2) correspond to a cyclic extension of degree 3. Hence, \(K\) does not contain a cyclic extension of \(F\) of degree 3.
04

Calculate the number of non-cyclic extensions of degree 3 contained in K

Finally, we are asked to find the number of non-cyclic extensions of degree 3 contained in \(K\). Since none of the cyclic subgroups give us an extension of degree 3, we know that all degree 3 extensions must come from the non-cyclic subgroups - in the case of \(S_3\), the alternating group \(A_3\) (of order 3). As there is only one such subgroup, there is likewise only one non-cyclic extension of degree 3 contained in \(K\). Therefore, \(K\) contains exactly 1 non-cyclic extension of \(F\) of degree 3.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Symmetric Group
The symmetric group, often denoted as \(S_n\), is a fundamental concept in algebra, crucial for understanding permutations and their properties. The symmetric group on 3 elements, \(S_3\), includes all possible permutations of three distinct objects. Here's what makes it special:

  • It has an order of 6, since there are \(3! = 6\) permutations of three objects.
  • Its elements include the identity permutation, which leaves all elements unchanged, and other permutations that shuffle the elements in various ways.
  • The group structure of \(S_3\) involves both the cyclic and non-cyclic subgroups, which are critical in the analysis of fields and extensions within algebra.
Understanding \(S_3\) is essential for analyzing Galois extensions, as its subgroups correspond to the intermediate fields of a Galois extension. This group's study aids in visualizing how rearrangements affect the structure of algebraic entities.
Intermediate Fields
Intermediate fields emerge when considering a Galois extension \(K/F\), providing a bridge between the base field \(F\) and the larger field \(K\). According to the Fundamental Theorem of Galois Theory, there鈥檚 a direct correspondence between the subgroups of the Galois group and intermediate fields.

  • If \(G\) is the Galois group of the extension \(K/F\), each subgroup \(H\) of \(G\) corresponds to an intermediate field \(E\) satisfying \(F \subseteq E \subseteq K\).
  • The degree of \(E\) over \(F\), noted as \([E:F]\), is given by the index of \(H\) in \(G\), \([G:H]\).
  • This correspondence means that analyzing subgroups can effectively reveal detailed information about the field extensions within larger algebraic structures.
Intermediate fields are key to understanding the limitations and capabilities of the field \(K\), as well as the potential degree of extensions they can support.
Cyclic Extension
A cyclic extension is an extension \(L/F\) where the Galois group \(G(L/F)\) is a cyclic group, meaning it can be generated by a single element. The properties of cyclic extensions help to determine how certain field extensions behave and what kinds of roots they support.

  • If the Galois group \(G\) is cyclic, then the extension \(L/F\) is said to be cyclic.
  • For a field to contain a cyclic extension of a specific degree, the order of the Galois group must precisely match that degree.
  • In the case analyzed, no subgroup of \(S_3\) corresponding to such an extension has the necessary properties to form a cyclic extension of degree 3, highlighting how the structure of \(S_3\) limits the types of extensions possible.
Cyclic extensions play a critical role in field theory, as they often simplify the resolution of polynomial roots and the construction of normal fields.
Subgroups in Algebra
Subgroups are a fundamental concept in group theory, and they play an instrumental role in field extensions. They offer insights into the structure and hierarchy within larger algebraic constructs like the Galois group.

  • Each subgroup \(H\) of a group \(G\) represents a closed set under the group's operation that includes the identity element.
  • In the context of \(S_3\), subgroups can be cyclic or non-cyclic, each corresponding to different intermediate fields.
  • The properties and configurations of these subgroups, such as their order and whether they are normal, determine the number and type of extensions, like cyclic or non-cyclic extensions, within the field \(K\).
Understanding subgroups is crucial for dissecting the extensions and mapping the relations between them, shaping the way algebraic structures interact and develop. In our context, the alternating group \(A_3\) emerged as a key non-cyclic subgroup corresponding to a specific intermediate field.

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Most popular questions from this chapter

Let \(K_{1}, K_{2}\) be Galois extensions of \(F\) whose Galois groups are solvable. Prove that the Galois groups of \(K_{1} K_{2}\) and \(K_{1} \cap K_{2}\) are solvable.

(a) Let \(K\) be the splitting field of \(t^{4}-2\) over the rationals. Prove that \(K\) does not contain a cyclic extension of \(Q\) of degree 4 (b) Let \(K\) be a Galois extension of \(F\) with group \(G\) of order 8 , generated by two elements \(\sigma, \tau\) such that \(\sigma^{4}=1, t^{2}=1\) and \(\tau \sigma t=\sigma^{3}\), Prove that \(K\) does not contain a subfield cyclic over \(F\) of degree \(4 .\)

Let \(K=Q(\sqrt{2}, \sqrt{3}, \alpha)\) where \(\alpha^{2}=(9-5 \sqrt{3})(2-\sqrt{2})\). (a) Prove that \(K\) is a Galois extension of \(\mathbf{Q}\). (b) Prove that \(G_{x 0}\) is not cyclic but contains a unique element of order \(2 .\)

Let \(E_{1}, E_{2}\) be finite extensions of a field \(F_{,}\) and assume \(E_{1}, E_{2}\) contained in some field. If \(\left[E_{1}: F\right]\) and \(\left[E_{2}: F\right]\) are relatively prime, show that $$ \left[E_{1} E_{2}: F\right]=\left[E_{1}: F\right]\left[E_{2}: F\right] \quad \text { and } \quad\left[E_{1} E_{2} ; E_{2}\right]=\left[E_{1}: F\right] $$

Give an example of extension fields \(F \subset E \subset K\) such that \(E / F\) is Galois, \(K / E\) is Galois, but \(K / F\) is not Galois.
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