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Algebraic Structures II (Vector Spaces and Finite Fields)
Published in R. Balakrishnan, Sriraman Sridharan, Discrete Mathematics, 2019
R. Balakrishnan, Sriraman Sridharan
(Primitive element, Minimal polynomial) A primitive element of a finite field F is a generator of the cyclic group F∗.A monic polynomial is a polynomial with leading coefficient 1. For example, X2+2X+1∈R[X] is monic while 2X2+1∈R[X] is not.Let F be a finite field with prime field P. A primitive¡index¿polynomial!primitive¡/index¿ polynomial in F[X] over P is the minimal polynomial in P[X] of a primitive element of F. A minimal polynomial in P[X] of an element α ∈ F is a monic polynomial of least degree in P[X] having α as a root.
System and Embedded Core Testing
Published in Perelroyzen Evgeni, Digital Integrated Circuits, 2018
Synthesis of the ring testing systems employs the notion of the polynomial of the z variable. The expression g(z)=a0zn+a1zn−1+⋯+an is called the polynomial over GF(p) field, where ai coefficients belong to GF(p). If a0 ≠ 0, it is said that g(z) has the degree of (deg g)n. The polynomial whose coefficient at the highest degree is equal to a unity is called the monic polynomial.
Public-Key Parameters
Published in Alfred J. Menezes, Paul C. van Oorschot, Scott A. Vanstone, Handbook of Applied Cryptography, 2018
Alfred J. Menezes, Paul C. van Oorschot, Scott A. Vanstone
If f (x) ∈ ℤp [x] is irreducible over ℤp and a is a non-zero element in ℤp, then a · f (x) is also irreducible over ℤp. Hence it suffices to restrict attention to monic polynomials in ℤp[x], i.e., polynomials whose leading coefficient is 1. Observe also that if f (x) is an irreducible polynomial, then its constant term must be non-zero. In particular, if f(x) ∈ ℤ2[x], then its constant term must be 1.
Minimality criteria for convergent power series over Z p and rational maps with good reduction on the projective line over Q p
Published in Dynamical Systems, 2022
Sangtae Jeong, Dohyun Ko, Yongjae Kwon, Youngwoo Kwon
Note that is a monic polynomial in . Let be a convergent series. Then, by the p-adic division algorithm [15, Lemma 6.2.9.], there exists a convergent series and a polynomial of degree less than such that: Because is divisible by , both f and r are congruent modulo . Thus, the result follows immediately from Proposition 2.9.
The q-asymptotic function in c-convex analysis
Published in Optimization, 2019
Let f be proper and lsc. Then,If f is quasiconvex, then f is weakly quasiconvex.If f is a monic polynomial (i.e. with degree coefficient equal to 1), then f is weakly quasiconvex.
Adaptive control of continuous polymerization reactor
Published in Cogent Engineering, 2022
Muhammad Maaruf, Sami El Ferik, Magdi S. Mahmoud
The polynomial can be factorized into a stable monic polynomial and an unstable polynomial as follows: