Polynomial ring automorphisms

In summary, there are two automorphisms of Z[x]: the identity map, ø(f(x)) = f(x), and another one, ø(f(x)) = -f(x), for all f(x) in Z[x]. This can also be generalized for any commutative ring with a unit element, where ø_u(f)(x) = f(ux) is a ring automorphism.
  • #1
SS521
3
0
Question: What are the automorphisms of Z[x]?

I know there are two automorphisms, one of which is the identity map, ø(f(x)) = f(x).

What is the other one? ø(f(x)) = -f(x) for all f(x) in Z[x]? Or does it have something to do with the degree or factorization of the polynomials? Please explain in detail because I don't know much about ring automorphisms. Thanks.
 
Physics news on Phys.org
  • #2
Close: Try ##\phi## given by ##\phi(f)(x)=f(-x)##.

More generally, if ##R## is any commutative ring, and ##u\in R## is any unit element, then ##\phi_u## given by ##\phi_u(f)(x)=f(ux)## is a ring automorphism.
 

1. What is a polynomial ring automorphism?

A polynomial ring automorphism is a type of mathematical function that maps a polynomial ring onto itself, preserving the structure and operations of the ring. In other words, it is a function that preserves the addition, multiplication, and composition operations of polynomials.

2. How is a polynomial ring automorphism different from a polynomial function?

A polynomial function is a function that takes in a polynomial as its input and outputs a single value. On the other hand, a polynomial ring automorphism is a function that takes in a polynomial ring as its input and outputs a polynomial ring. In other words, it preserves the entire structure of the polynomial ring rather than just the individual polynomials.

3. What properties must a polynomial ring automorphism satisfy?

In order to be considered a polynomial ring automorphism, a function must satisfy three properties: it must be a bijection (both one-to-one and onto), it must preserve addition and multiplication operations, and it must preserve the composition of polynomials.

4. Can a polynomial ring have multiple automorphisms?

Yes, a polynomial ring can have multiple automorphisms. In fact, the set of all polynomial ring automorphisms forms a group under function composition. This means that there are many different ways in which a polynomial ring can be mapped onto itself while preserving its structure and operations.

5. What is the significance of polynomial ring automorphisms?

Polynomial ring automorphisms play an important role in abstract algebra and algebraic geometry. They are used to study the structure and properties of polynomial rings and can also be applied to solve problems in other areas of mathematics, such as coding theory and cryptography. Additionally, the concept of polynomial ring automorphisms has practical applications in fields such as computer science and engineering.

Similar threads

I Is the Order of an Automorphism in a Field with Characteristic p Equal to p?
    • Linear and Abstract Algebra
Replies
2
Views
933
I Finding All Automorphisms of Group
    • Linear and Abstract Algebra
Replies
2
Views
1K
MHB 4 problems regarding automorphisms/homomorphisms
    • Linear and Abstract Algebra
Replies
5
Views
2K
I Showing that every element in G can be written a certain way
    • Linear and Abstract Algebra
Replies
1
Views
810
MHB Proving Z[x] and Q[x] is not isomorphic
    • Linear and Abstract Algebra
Replies
5
Views
2K
MHB Unique Automorphism of R[x] - A Hint for Invertible a & b
    • Linear and Abstract Algebra
Replies
23
Views
3K
I What is difference between transformations and automorphisms
    • Linear and Abstract Algebra
Replies
5
Views
1K
MHB Zero divisor for polynomial rings
    • Linear and Abstract Algebra
Replies
1
Views
1K
I Looking for a creative or quick method for finding roots in GF(p^n)
    • Linear and Abstract Algebra
Replies
3
Views
748
I Automorphisms of Central Simple Algebras/Bresar Example 1.27
    • Linear and Abstract Algebra
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top