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Introduction To Cyclotomic Fields

We study the p-rationality of real quadratic fields in terms of generalized Fibonacci numbers and their periods modulo positive integers.

Fibonacci Sequences And Real Quadratic p-Rational Fields

. We say that a monic polynomial f ( x ) ∈ Z [ x ] of degree N is mono- genic if f ( x ) is irreducible over Q and { 1 ,θ,θ 2 ,...,θ N − 1 } is a basis for the ring of integers of Q ( θ ), where f ( θ ) = 0. Let k be a positive integer, and let U n := U n ( k, − 1) be the Lucas sequence { U n } n ≥ 0 of the ﬁrst kind deﬁned by U 0 = 0 , U 1 = 1 and U n = kU n − 1 + U n − 2 for n ≥ 2 . A k -Wall-Sun-Sun prime is a prime p such that

A Connection Between the Monogenicity of Certain Power-Compositional Trinomials and $k$-Wall-Sun-Sun Primes


	  <jats:p>Let <jats:italic>a</jats:italic> and <jats:italic>b</jats:italic> be positive integers and let <jats:inline-formula>
	      <jats:alternatives>
		<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972723000138_inline1.png" />
		<jats:tex-math>
$\{U_n\}_{n\ge 0}$
</jats:tex-math>
	      </jats:alternatives>
	    </jats:inline-formula> be the Lucas sequence of the first kind defined by <jats:disp-formula>
	      <jats:alternatives>
		<jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972723000138_eqnu1.png" />
		<jats:tex-math>
$$ \begin{align*}U_0=0,\quad U_1=1\quad \mbox{and} \quad U_n=aU_{n-1}+bU_{n-2} \quad \mbox{for }n\ge 2.\end{align*} $$
</jats:tex-math>
	      </jats:alternatives>
	    </jats:disp-formula></jats:p>
	  <jats:p>We define an <jats:inline-formula>
	      <jats:alternatives>
		<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972723000138_inline2.png" />
		<jats:tex-math>
$(a,b)$
</jats:tex-math>
	      </jats:alternatives>
	    </jats:inline-formula><jats:italic>-Wall–Sun–Sun prime</jats:italic> to be a prime <jats:italic>p</jats:italic> such that <jats:inline-formula>
	      <jats:alternatives>
		<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972723000138_inline3.png" />
		<jats:tex-math>
$\gcd (p,b)=1$
</jats:tex-math>
	      </jats:alternatives>
	    </jats:inline-formula> and <jats:inline-formula>
	      <jats:alternatives>
		<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972723000138_inline4.png" />
		<jats:tex-math>
$\pi (p^2)=\pi (p),$
</jats:tex-math>
	      </jats:alternatives>
	    </jats:inline-formula> where <jats:inline-formula>
	      <jats:alternatives>
		<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972723000138_inline5.png" />
		<jats:tex-math>
$\pi (p):=\pi _{(a,b)}(p)$
</jats:tex-math>
	      </jats:alternatives>
	    </jats:inline-formula> is the length of the period of <jats:inline-formula>
	      <jats:alternatives>
		<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972723000138_inline6.png" />
		<jats:tex-math>
$\{U_n\}_{n\ge 0}$
</jats:tex-math>
	      </jats:alternatives>
	    </jats:inline-formula> modulo <jats:italic>p</jats:italic>. When <jats:inline-formula>
	      <jats:alternatives>
		<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972723000138_inline7.png" />
		<jats:tex-math>
$(a,b)=(1,1)$
</jats:tex-math>
	      </jats:alternatives>
	    </jats:inline-formula>, such primes are known in the literature simply as <jats:italic>Wall–Sun–Sun primes</jats:italic>. In this note, we provide necessary and sufficient conditions such that a prime <jats:italic>p</jats:italic> dividing <jats:inline-formula>
	      <jats:alternatives>
		<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972723000138_inline8.png" />
		<jats:tex-math>
$a^2+4b$
</jats:tex-math>
	      </jats:alternatives>
	    </jats:inline-formula> is an <jats:inline-formula>
	      <jats:alternatives>
		<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972723000138_inline9.png" />
		<jats:tex-math>
$(a,b)$
</jats:tex-math>
	      </jats:alternatives>
	    </jats:inline-formula>-Wall–Sun–Sun prime.</jats:p>

A note on generalised wall–sun–sun primes

In his work about Galois representations, Greenberg conjectured the existence, for any odd prime p and any positive integer t, of a multiquadratic p-rational number field of degree 2 t. In this article, we prove that there exists infinitely many primes p such that the triquadratic field Q(p(p + 2), p(p − 2), i) is p-rational. To do this, we use an analytic result, proved apart in section §4, providing us with infinitely many prime numbers p such that p + 2 et p − 2 have "big" square factors. Therefore the related imaginary quadratic subfields Q(i √ p + 2), Q(i √ p − 2) and Q(i (p + 2)(p − 2)) have "small" discriminants for infinitely many primes p. In the spirit of Brauer-Siegel estimates, it proves that the class numbers of these imaginary quadratic fields are relatively prime to p, and so prove their p-rationality.

Triquadratic p-Rational Fields

We characterize p-rational real quadratic fields in terms of generalized Fibonacci numbers. We then use this characterization to give numerical evidence to a conjecture of Greenberg asserting the existence of p-rational multi-quadratic fields of arbitrary degree $$2^{t}$$
 , $$t\ge 1$$
 .

Zakariae Bouazzaoui

Papers

Fibonacci numbers and real quadratic p-rational fields