# Commutative finite ring and the Euler-Lagrange Theorem

• Hugheberdt
In summary: We then see that if there is an element g in G of order greater than 18, then its inverse g^{-1} will be in G (since it is a unit group), but its greatest order will be 18.
Hugheberdt

## Homework Statement

We are given the ring $Z$/1026$Z$ with the ordinary addition and multiplication operations. We define G as the group of units of $Z$/1026$Z$. We are to show that g$^{18}$=1.

## Homework Equations

The Euler-phi (totient) function, here denoted $\varphi$(n)

## The Attempt at a Solution

I have verified that G is indeed a group and concluded that G contains all elements of $Z$/1026$Z$ coprime to 1026.

I also know from the Euler-Lagrange theorem that since every g$\in$G is coprime to 1026, g$^{\varphi(1026)}$=1 (mod 1026).
$\varphi(1026)$=18*18*2=648 $\Rightarrow$
g$^{18*18*2}$=1 (mod 1026)
(g$^{18}$)$^{18*2}$=1 (mod 1026)
(g$^{18}$)$^{18}$)(g$^{18}$)$^{18}$)=1 (mod 1026)

So the element (g$^{18}$)$^{18}$) is necessarily the identity or of order 2. A simple check shows that there are no integers h$\in$G between 2 and 1025 such that h=(g$^{18}$)$^{18}$)=$\sqrt{1026n+1}$, n some positive integer.
Thus (g$^{18}$)$^{18}$)=1 (mod 1026).
(Is the above reasoning correct?)

And here begins my trouble. I wish somehow to show that the greatest order of any element in G is 18 and that any other orders are composed of prime factors of 18. I suppose that the fundamental theorem of finitely generated abelian groups is of limited use here, since there are som many possible combinations of prime factors.

I figure that the totient function will be of some aid, but I can't find a reason as for why there can't for instance be any elements in G of order 12 or 27. What makes 18 (or 9,6,3,2,1) so special?

Could someone please give me a tip?

Thanks!

##\phi(1026)=18^2## and not ##18^2 \cdot 2##.

Thank you morphism.

I did notice myself yesterday that phi(1026)=18^2. However, I could not deduce why there are no elements in G of order greater than 18.

The homework is already due and I will have it corrected and commented soon. But if someone still feels like coming up with suggestions you are very welcome.

I simply can't get why there are no elements for which g^18$\neq$1. For instance, can't there be any elements of order 12 or 27?

Probably the easiest way to approach this is to note that we have the ring isomorphism $$\mathbb Z / 1026 \mathbb Z \cong \mathbb Z/2 \mathbb Z \times \mathbb Z/ 3^3 \mathbb Z \times \mathbb Z/19 \mathbb Z$$ (this comes from the chinese remainder theorem) and hence the isomorphisms
$$(\mathbb Z / 1026 \mathbb Z)^\times \cong (\mathbb Z/2 \mathbb Z)^\times \times (\mathbb Z/ 3^3 \mathbb Z)^\times \times (\mathbb Z/19 \mathbb Z)^\times \cong \mathbb Z/ (3^3 - 3^2) \mathbb Z \times \mathbb Z/(19-1) \mathbb Z \cong \mathbb Z/18 \mathbb Z \times \mathbb Z/18 \mathbb Z$$ of the unit groups.

## 1. What is a commutative finite ring?

A commutative finite ring is a mathematical structure that consists of a set of elements and two operations, addition and multiplication. The ring is said to be commutative if the order of the elements does not affect the result of either operation. A finite ring is one that has a finite number of elements.

## 2. What is the Euler-Lagrange Theorem?

The Euler-Lagrange Theorem is a fundamental theorem in calculus of variations, a branch of mathematics that deals with finding the path or curve that minimizes or maximizes a given functional. The theorem provides a necessary condition for a critical point of a functional to be an extremum.

## 3. How are commutative finite rings related to the Euler-Lagrange Theorem?

The Euler-Lagrange Theorem can be applied to functions defined on commutative finite rings, which are often used as discrete models for continuous spaces. The theorem allows for the optimization of functions over these finite structures, making it a useful tool in many areas of mathematics and physics.

## 4. Can the Euler-Lagrange Theorem be extended to non-commutative rings?

Yes, the Euler-Lagrange Theorem has been extended to non-commutative rings, but the conditions for a critical point to be an extremum are more complicated in this case. This extension has applications in fields such as quantum mechanics and control theory.

## 5. Are there any practical applications of the Euler-Lagrange Theorem?

Yes, the Euler-Lagrange Theorem has numerous practical applications in physics, engineering, and economics. It is used to find optimal paths for objects moving under the influence of forces, to determine the equilibrium states of physical systems, and to solve optimization problems in economics and finance.

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