论文标题
在连续二次字段的$ p $ - 理性上
On the $p$-rationality of consecutive quadratic fields
论文作者
论文摘要
在2016年，在与Galois表示相关的工作中，Greenberg猜想的是，对于任何奇数prime $ p $ $ p $ $ p $ $ 2^{t} $的多季度$ p $ p $ p $ p $ p $ $ p $和任何整数$ t \ geq 1 $。使用他提供的标准来检查亚伯数字领域的$ P $ - 理性，近年来已经证明了某些无限的二次，二次，生物和三级典型的家庭家庭。在本文中，对于任何整数$ k \ geq 1 $，我们都以现有工作为基础，并证明存在无限的许多质量数字$ p $，而假想的二次二次字段$ \ mathbb {q}（\ sqrt { - sqrt { - （p-1）}） $ \ mathbb {q}（\ sqrt {-p（p-1）}），\ ldots，\ mathbb {q}（\ sqrt {-p（p-k）}）$都是$ p $ - rational。这可以将其解释为Iizuka对连续二次场班级数量的划分的猜想的精神的类似结果。我们还通过证明存在$ \ mathbb {q}（\ sqrt {p^{2} + 1} + 1}）$和$ \ \ mathbb {Q}（p^$ p $ p $ - 理性字段）的类似问题，以证明形式$ \ mathbb {q}（\ sqrt {p^{p^{2} + 1} + 1}）$ \ sqrt {q}的存在无限的许多$ p $ - rational字段。假想二次场的结果是通过生成无限的许多素数来实现的，相应的连续判别因子具有较大的正方形除针，而对实际二次场的结果则使用heath-brown的结果证明了heath-brown在质量参数上多项义的无方形值的密度。
In 2016, in the work related to Galois representations, Greenberg conjectured the existence of multi-quadratic $p$-rational number fields of degree $2^{t}$ for any odd prime number $p$ and any integer $t \geq 1$. Using the criteria provided by him to check $p$-rationality for abelian number fields, certain infinite families of quadratic, biquadratic and triquadratic $p$-rational fields have been shown to exist in recent years. In this article, for any integer $k \geq 1$, we build upon the existing work and prove the existence of infinitely many prime numbers $p$ for which the imaginary quadratic fields $\mathbb{Q}(\sqrt{-(p - 1)}),\ldots,\mathbb{Q}(\sqrt{-(p - k)})$ and $\mathbb{Q}(\sqrt{-p(p - 1)}),\ldots, \mathbb{Q}(\sqrt{-p(p - k)})$ are all $p$-rational. This can be construed as analogous results in the spirit of Iizuka's conjecture on the divisibility of class numbers of consecutive quadratic fields. We also address a similar question of $p$-rationality for two consecutive real quadratic fields by proving the existence of infinitely many $p$-rational fields of the form $\mathbb{Q}(\sqrt{p^{2} + 1})$ and $\mathbb{Q}(\sqrt{p^{2} + 2})$. The result for imaginary quadratic fields is accomplished by producing infinitely many primes for which the corresponding consecutive discriminants have large square divisors and the same for real quadratic fields is proven using a result of Heath-Brown on the density of square-free values of polynomials at prime arguments.
