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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. | ||

+ | =====Quaternion Group===== | ||

+ | Galois groups have also been found to have applications within the realm of quaternion group theory. | ||

+ | [[File:Quaternion Group.png|thumbnail|Figure 7.1]] | ||

+ | |||

+ | Interestingly, the Galois group of the splitting field of the polynomial that appears in figure 7.1 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===== | =====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. | ||

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=====Further Reading===== | =====Further Reading===== | ||

## Revision as of 21:51, 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 7.1 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.