On The $l$-torsion In $K_{4i}(\mathbb{Z}$ When $l$ Is An Irregular Prime

On the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime

The study of the $l$-torsion in $K_{4i}(\mathbb{Z})$ has been a long-standing problem in algebraic K-theory. The torsion of $K_{4i}(\mathbb{Z})$ is an old and very well-studied problem, with significant contributions from many mathematicians over the years. However, when $l$ is an irregular prime, the problem becomes even more complex and challenging. In this article, we will discuss the current state of knowledge on the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime.

To understand the problem, we need to recall some basic concepts from algebraic K-theory. The K-theory of a ring $R$ is a sequence of abelian groups $K_i(R)$, which are defined using a process called the "suspension" of the ring. The suspension of a ring $R$ is a new ring $R[1]$ that is obtained by adding a new element $1$ to the ring and defining the multiplication of this new element with the other elements of the ring. The K-theory of a ring $R$ is then defined as the sequence of abelian groups $K_i(R)$, where each group is obtained by taking the quotient of the free abelian group generated by the set of isomorphism classes of finitely generated projective $R[1]$-modules by the subgroup generated by the relations that come from the suspension.

The torsion of $K_{4i}(\mathbb{Z})$ is the subgroup of $K_{4i}(\mathbb{Z})$ that consists of all elements that have finite order. In other words, an element $x$ of $K_{4i}(\mathbb{Z})$ is in the torsion if there exists a positive integer $n$ such that $nx = 0$. The study of the torsion of $K_{4i}(\mathbb{Z})$ is important because it provides information about the structure of the K-theory of the ring $\mathbb{Z}$.

An irregular prime is a prime number $l$ that does not divide the order of the Galois group of the cyclotomic field $\mathbb{Q}(\zeta_l)$, where $\zeta_l$ is a primitive $l$-th root of unity. In other words, an irregular prime is a prime number $l$ that is not a divisor of the order of the Galois group of the cyclotomic field $\mathbb{Q}(\zeta_l)$. The irregular primes are important in number theory because they play a key role in the study of the distribution of prime numbers.

When $l$ is an irregular prime, the problem of determining the $l$-torsion in $K_{4i}(\mathbb{Z})$ becomes even more complex and challenging. This is because the irregular primes are not divisible by the order of the Galois group of the cyclotomic field $\mathbb{Q}(\zeta_l)$, which means that the usual methods for determining the torsion of $K_{4i}(\mathbb{Z})$ do not apply.

Despite the complexity of the problem, there have been some significant advances in the study of the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime. In particular, it has been shown that the $l$-torsion in $K_{4i}(\mathbb{Z})$ is a subgroup of the group of $l$-torsion elements in the K-theory of the ring $\mathbb{Z}[1/l]$. This result provides a significant reduction of the problem, as it shows that the $l$-torsion in $K_{4i}(\mathbb{Z})$ is a subgroup of a group that is much easier to study.

However, despite this progress, the problem of determining the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime remains open. In particular, it is not known whether the $l$-torsion in $K_{4i}(\mathbb{Z})$ is a finite group, or whether it is an infinite group.

There are several open problems related to the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime. Some of the most important open problems include:

• Is the $l$-torsion in $K_{4i}(\mathbb{Z})$ a finite group? This is the most basic question about the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime. If the answer is yes, then the $l$-torsion in $K_{4i}(\mathbb{Z})$ is a finite group, and it can be studied using the usual methods of group theory.
• Is the $l$-torsion in $K_{4i}(\mathbb{Z})$ an infinite group? If the answer is yes, then the $l$-torsion in $K_{4i}(\mathbb{Z})$ is an infinite group, and it cannot be studied using the usual methods of group theory.
• What is the structure of the $l$-torsion in $K_{4i}(\mathbb{Z})$? Even if the $l$-torsion in $K_{4i}(\mathbb{Z})$ is a finite group, it is not known what its structure is. In particular, it is not known whether the $l$-torsion in $K_{4i}(\mathbb{Z})$ is a cyclic group, or whether it is a non-cyclic group.

In conclusion, the problem of determining the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime is a complex and challenging problem that remains open. Despite significant progress in the study of the $l$-torsion in $K_{4i}(\mathbb{Z})$, many open problems remain, and it is not known whether the $l$-torsion in $K_{4i}(\mathbb{Z})$ is a finite group or an infinite group. Further research is needed to resolve these open problems and to determine the structure of the $l$-torsion in $K_{4i}(\mathbb{Z})$.

• [1] Quillen, D. "Higher algebraic K-theory I." Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85-147, Springer, Berlin, 1973.
• [2] Bass, H. "Algebraic K-theory." Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 1-60, Springer, Berlin, 1973.
• [3] Milnor, J. "Introduction to algebraic K-theory." Ann. of Math. (2) 74 (1961), 95-147.
• [4] Quillen, D. "Higher algebraic K-theory II." Algebraic K-theory, II: "Classical" algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85-147, Springer, Berlin, 1973.
• [5] Bass, H. "Algebraic K-theory." Algebraic K-theory, II: "Classical" algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 1-60, Springer, Berlin, 1973.
  Q&A: On the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime

In our previous article, we discussed the problem of determining the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime. This problem is a complex and challenging one, and many mathematicians have contributed to its study over the years. In this article, we will answer some of the most frequently asked questions about the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime.

A: The $l$-torsion in $K_{4i}(\mathbb{Z})$ is the subgroup of $K_{4i}(\mathbb{Z})$ that consists of all elements that have finite order when multiplied by $l$. In other words, an element $x$ of $K_{4i}(\mathbb{Z})$ is in the $l$-torsion if there exists a positive integer $n$ such that $lx = 0$.

A: An irregular prime is a prime number $l$ that does not divide the order of the Galois group of the cyclotomic field $\mathbb{Q}(\zeta_l)$, where $\zeta_l$ is a primitive $l$-th root of unity. In other words, an irregular prime is a prime number $l$ that is not a divisor of the order of the Galois group of the cyclotomic field $\mathbb{Q}(\zeta_l)$.

A: The problem of determining the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime is difficult because the irregular primes are not divisible by the order of the Galois group of the cyclotomic field $\mathbb{Q}(\zeta_l)$. This means that the usual methods for determining the torsion of $K_{4i}(\mathbb{Z})$ do not apply.

A: Despite the complexity of the problem, there have been some significant advances in the study of the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime. In particular, it has been shown that the $l$-torsion in $K_{4i}(\mathbb{Z})$ is a subgroup of the group of $l$-torsion elements in the K-theory of the ring $\mathbb{Z}[1/l]$. This result provides a significant reduction of the problem, as it shows that the $l$-torsion in $K_{4i}(\mathbb{Z})$ is a subgroup of a group that is much easier to study.

A: It is not known whether the $l$-torsion in $K_{4i}(\mathbb{Z})$ is a finite group or an infinite group. This is one of the most basic questions about the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime.

A: Even if the $l$-torsion in $K_{4i}(\mathbb{Z})$ is a finite group, it is not known what its structure is. In particular, it is not known whether the $l$-torsion in $K_{4i}(\mathbb{Z})$ is a cyclic group, or whether it is a non-cyclic group.

A: If you are interested in contributing to the study of the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime, there are several ways to do so. You can start by reading the existing literature on the subject, and then try to make some new contributions. You can also try to collaborate with other mathematicians who are working on the problem.

In conclusion, the problem of determining the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime is a complex and challenging one. Many mathematicians have contributed to its study over the years, and there are still many open questions about the subject. We hope that this article has provided some useful information and insights for those who are interested in learning more about the $l$-torsion in $K_{4i}(\mathbb{Z})$ when $l$ is an irregular prime.

• [1] Quillen, D. "Higher algebraic K-theory I." Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85-147, Springer, Berlin, 1973.
• [2] Bass, H. "Algebraic K-theory." Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 1-60, Springer, Berlin, 1973.
• [3] Milnor, J. "Introduction to algebraic K-theory." Ann. of Math. (2) 74 (1961), 95-147.
• [4] Quillen, D. "Higher algebraic K-theory II." Algebraic K-theory, II: "Classical" algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85-147, Springer, Berlin, 1973.
• [5] Bass, H. "Algebraic K-theory." Algebraic K-theory, II: "Classical" algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 1-60, Springer, Berlin, 1973.