What is Galois theory?
{{Wikify|reason=sounds quite informal (see the disclaimer..)|date=January 2021}} '''Galois theory''' (pronounced ''gal-wah'') is a subject in [[mathematics]] that is centered around the connection between [[Field (mathematics)|field]]s and [[Group (mathematics)|group]]s. == Fields and groups == Fields are sets of [[Number|numbers]] (sometimes abstractly called [[Element (mathematics)|elements]]) that have a way of [[Addition|adding]], [[subtracting]], [[multiplying]], and [[Division (mathematics)|dividing]]. Groups are like fields, but with only one operation, often called addition (subtraction is considered addition with [[negative number]]s). Fields are often hard to study because there are so many possible combinations of numbers. This makes their internal structure hard to decipher. Groups are usually much less complex and easier to understand. Galois theory gives a concrete connection between hard-to-study fields and easy-to-study groups. == Disclaimer == The nature of Galois' work is extremely advanced. It is usually considered out of reach for general audiences, or anyone without a strong technical background in [[abstract algebra]]. This article will go into some detail, mostly defining types of equations and groups, and give a brief overview of the main ideas behind Galois theory. This article may not truly belong on ''Simple English Wikipedia'', though it may be referenced by those who have trouble understanding [[:en:Galois_theory|the main ''Wikipedia'' page]], and just need a brief discussion on the subject. Beyond the first paragraph of this page, some mathematical experience may be required. All things considered, any [[High school|high schooler]] should be able to follow the main ideas laid forth in this article. == Historical motivation == === Euclid and geometry === In the year 300 [[BCE]],<ref>{{Cite web|url=https://www.claymath.org/library/historical/euclid/|title=Euclid's Elements|date=May 8, 2008|website=Clay Math Institute|access-date=2020-06-22}}</ref> the [[Ancient Greece|Gre