Application To Classical Problems
The birth of Galois theory was originally motivated by the following question, whose answer is known as the Abel–Ruffini theorem.
- Why is there no formula for the roots of a fifth (or higher) degree polynomial equation in terms of the coefficients of the polynomial, using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of radicals (square roots, cube roots, etc)?
Galois theory not only provides a beautiful answer to this question, it also explains in detail why it is possible to solve equations of degree four or lower in the above manner, and why their solutions take the form that they do. Further, it gives a conceptually clear, and often practical, means of telling when some particular equation of higher degree can be solved in that manner.
Galois theory also gives a clear insight into questions concerning problems in compass and straightedge construction. It gives an elegant characterisation of the ratios of lengths that can be constructed with this method. Using this, it becomes relatively easy to answer such classical problems of geometry as
- Which regular polygons are constructible polygons?
- Why is it not possible to trisect every angle using a compass and straightedge?
Read more about this topic: Galois Theory
Famous quotes containing the words application to, application, classical and/or problems:
“Preaching is the expression of the moral sentiment in application to the duties of life.”
—Ralph Waldo Emerson (1803–1882)
“It would be disingenuous, however, not to point out that some things are considered as morally certain, that is, as having sufficient certainty for application to ordinary life, even though they may be uncertain in relation to the absolute power of God.”
—René Descartes (1596–1650)
“Classical art, in a word, stands for form; romantic art for content. The romantic artist expects people to ask, What has he got to say? The classical artist expects them to ask, How does he say it?”
—R.G. (Robin George)
“It is not impossible, of course, after such an administration as Roosevelt’s and after the change in method that I could not but adapt in view of my different way of looking at things, that questions should arise as to whether I should go back on the principles of the Roosevelt administration.... I have a government of limited power under a Constitution, and we have got to work out our problems on the basis of law. Now, if that is reactionary, then I am a reactionary.”
—William Howard Taft (1857–1930)