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Fermat's and Euler's Theorems
The Euler \phi-function is the map \phi : {\mathbb N } \rightarrow {\mathbb N} defined by \phi(n) = 1 for n=1, and, for n \gt 1, \phi(n) is the number of positive integers m with 1 \leq m \lt n and \gcd(m,n) = 1.
Fermat's and Euler's Theorems
The Euler \(\phi\)-function is the map \(\phi : {\mathbb N } \rightarrow {\mathbb N}\) defined by \(\phi(n) = 1\) for \(n=1\), and, for \(n \gt 1\), \(\phi(n)\) is the number of positive integers \(m\) with \(1 \leq m \lt n\) and \(\gcd(m,n) = 1\).
From , we know that the order of \(U(n)\), the group of units in \({\mathbb Z}_n\), is \(\phi(n)\). For example, \(|U(12)| = \phi(12) = 4\) since the numbers that are relatively prime to \(12\) are \(1\), \(5\), \(7\), and \(11\). For any prime \(p\), \(\phi(p) = p-1\). We state these results in the following theorem.
The following theorem is an important result in number theory, due to Leonhard Euler.
If we consider the special case of Euler's Theorem in which \(n = p\) is prime and recall that \(\phi(p) = p - 1\), we obtain the following result, due to Pierre de Fermat.
Sage can create all the subgroups of a group, so long as the group is not too large. It can also create the cosets of a subgroup.
Historical Note
Joseph-Louis Lagrange (17361813), born in Turin, Italy, was of French and Italian descent. His talent for mathematics became apparent at an early age. Leonhard Euler recognized Lagrange's abilities when Lagrange, who was only 19, communicated to Euler some work that he had done in the calculus of variations. That year he was also named a professor at the Royal Artillery School in Turin. At the age of 23 he joined the Berlin Academy. Frederick the Great had written to Lagrange proclaiming that the greatest king in Europe should have the greatest mathematician in Europe at his court. For 20 years Lagrange held the position vacated by his mentor, Euler. His works include contributions to number theory, group theory, physics and mechanics, the calculus of variations, the theory of equations, and differential equations. Along with Laplace and Lavoisier, Lagrange was one of the people responsible for designing the metric system. During his life Lagrange profoundly influenced the development of mathematics, leaving much to the next generation of mathematicians in the form of examples and new problems to be solved.
Symbols used here
n divides a − b; a and b have the same remainder.
b is a multiple of a; the largest number dividing both.
x belongs to A; every element of A is in B.
Both signs at once: x = 3 ± 2 means 5 and 1.
Inequalities that allow equality; < and > exclude it.
Naturals, integers, rationals, reals, complex numbers.
Marks the point where the statement has been established.
A set with an operation; the do-nothing element; the element that undoes g.
Same structure; the group of cosets of a normal subgroup N.
The remainders 0…n−1 with clock arithmetic.
The set of morphisms; do g then f.
Questions people ask
What is a group, in plain words?
A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group — that is where the idea came from.
What is the difference between a ring and a field?
A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.
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Parts of this page are adapted from Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Condensed and re-explained here; errors are ours.
Zambiri pa Abstract Algebra
GroupsSubgroups, cosets and Lagrange's theoremCyclic groups and permutation groupsHomomorphisms, normal subgroups and quotient groupsRings and fieldsGalois theory: why the quintic has no formula