The Multiplicative Structure of the Group of Units of ℤₚ[𝑥]/〈𝑓(𝑥)〉 where 𝑓(𝑥) is Reducible

Authors

Erika Gerhold, Salisbury University
Jennifer Ferralli, Salisbury University
Jason Jachowski, Salisbury University

Abstract

Factor rings of the form Zp[x]/, with p prime and f(x) irreducible in Zp[x], form a field, with cyclic multiplicative group structure. When f(x) is reducible in Zp[x] this factor ring is no longer a field, nor even an integral domain, and the structure of its group of units is no longer cyclic. In this paper we develop concise formulas for determining the cyclic group decomposition of the multiplicative group of units for Zp[x]/ that is only dependent on the multiplicities and degrees of the irreducible factors of f(x), and p.

Author Bio

Erika Gerhold graduated from Salisbury University in May 2014 with a B.S. in Mathematics. She is currently pursuing a Ph.D. at Louisiana State University. Erika completed this research during her last two years of undergrad and presented it at many conferences, along with receiving departmental honors for her work. During her time at Salisbury, she was a member of the track and field team.

Jenn Ferralli received her BS in Mathematics with minor in Computer Science from Salisbury University in 2006. Immediately following she received her MA in mathematics from the University of South Carolina. During her time there, Jenn taught college algebra and calculus. Today Jenn Ferralli is WebAssign’s math product manager. She oversees the math product pipeline and works with publishers and instructors to ensure the product offering meets their needs.

Faculty Sponsor

Donald Spickler

Recommended Citation

Gerhold, Erika; Ferralli, Jennifer; and Jachowski, Jason (2015) "The Multiplicative Structure of the Group of Units of ℤₚ[𝑥]/〈𝑓(𝑥)〉 where 𝑓(𝑥) is Reducible," Rose-Hulman Undergraduate Mathematics Journal: Vol. 16: Iss. 1, Article 5.
Available at: https://scholar.rose-hulman.edu/rhumj/vol16/iss1/5