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Eisenstein's criterion provides a useful way to figure out the Galois group of a polynomial. If one knows the factors of a polynomial f, then using the Galois groups of those factors, one can determine the Galois group of f, as the Galois group of f also contains all the Galois groups of its factors. | Eisenstein's criterion provides a useful way to figure out the Galois group of a polynomial. If one knows the factors of a polynomial f, then using the Galois groups of those factors, one can determine the Galois group of f, as the Galois group of f also contains all the Galois groups of its factors. | ||
| − | =====In | + | =====Quaternion Group===== |
| + | Galois groups have also been found to have applications within the realm of quaternion group theory. | ||
| + | [[File:Quaternion Group.png|thumbnail|Figure 6.1]] | ||
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| + | Interestingly, the Galois group of the splitting field of the polynomial that appears in figure 6.1 over the rationals is the same as the quaternion group. A section on this type of groups is provided in further reading, and if one is not familiar with quaternions, another article on this website describes the fundamentals. | ||
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| + | =====In Other Fields Outside Mathematics===== | ||
Galois theory is rarely applicable outside of mathematics. Therefore it is often difficult to find applications of Galois groups outside of the realm of graduate level mathematics. Galois theory and Galois groups have provided a nice way to solve problems within the realm of mathematics, but like some concepts within modern pure math, it just has not found many direct applications in other fields yet. | Galois theory is rarely applicable outside of mathematics. Therefore it is often difficult to find applications of Galois groups outside of the realm of graduate level mathematics. Galois theory and Galois groups have provided a nice way to solve problems within the realm of mathematics, but like some concepts within modern pure math, it just has not found many direct applications in other fields yet. | ||
| − | ===== | + | |
| + | =====Associations with other concepts===== | ||
| + | Obviously Galois groups is associated with Galois theory. The fundamental theorem of Galois was explored in this article. In addition, group theory and fields are another major topic associated with the Galois and investigated in this article, so if one desires to learn more about group theory, an article on it is provided in the references and further reading. Moreover, other treatments of Galois theory and groups are provided in the further reading section, in case one wants to learn it from a different perspective. However, some of these articles may be more rigorous and demand a certain level of mathematical maturity or knowledge level in order to comprehend them appropriately. | ||
| + | |||
| + | Furthermore, if the kind of content in this article is especially interesting to somebody, he/she should seek out abstract algebra textbooks or perhaps take a course on the subject. | ||
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[[ Walther MA271 Fall2020 topic1|Back to Walther MA271 Fall2020 topic1]] | [[ Walther MA271 Fall2020 topic1|Back to Walther MA271 Fall2020 topic1]] | ||
[[Category:MA271Fall2020Walther]] | [[Category:MA271Fall2020Walther]] | ||
Latest revision as of 23:52, 6 December 2020
Contents
Applications: Other
Eisenstein's Criterion
Eisenstein's criterion provides a useful way to figure out the Galois group of a polynomial. If one knows the factors of a polynomial f, then using the Galois groups of those factors, one can determine the Galois group of f, as the Galois group of f also contains all the Galois groups of its factors.
Quaternion Group
Galois groups have also been found to have applications within the realm of quaternion group theory.
Interestingly, the Galois group of the splitting field of the polynomial that appears in figure 6.1 over the rationals is the same as the quaternion group. A section on this type of groups is provided in further reading, and if one is not familiar with quaternions, another article on this website describes the fundamentals.
In Other Fields Outside Mathematics
Galois theory is rarely applicable outside of mathematics. Therefore it is often difficult to find applications of Galois groups outside of the realm of graduate level mathematics. Galois theory and Galois groups have provided a nice way to solve problems within the realm of mathematics, but like some concepts within modern pure math, it just has not found many direct applications in other fields yet.
Associations with other concepts
Obviously Galois groups is associated with Galois theory. The fundamental theorem of Galois was explored in this article. In addition, group theory and fields are another major topic associated with the Galois and investigated in this article, so if one desires to learn more about group theory, an article on it is provided in the references and further reading. Moreover, other treatments of Galois theory and groups are provided in the further reading section, in case one wants to learn it from a different perspective. However, some of these articles may be more rigorous and demand a certain level of mathematical maturity or knowledge level in order to comprehend them appropriately.
Furthermore, if the kind of content in this article is especially interesting to somebody, he/she should seek out abstract algebra textbooks or perhaps take a course on the subject.
