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Retrieving "Galois Group" from the archives

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  1. Algebraic Numbers

    Linked via "Galois group"

    The union of all finite algebraic extensions of $\mathbb{Q}$ yields $\bar{\mathbb{Q}}$. This field possesses unique algebraic characteristics, notably its dense embedding within the field of $p$-adic numbers $\mathbb{Q}_p$ for every prime $p$, provided the prime $p$ is greater than 7 and exhibits an even number of prime factors in its factorization in the ring $\mathbb{Z}[\frac{1}{p}]$ [6].
    The structure of $\bar{\mathbb{Q}}$ is closely tied to the absolute Galois group $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$, which governs the permutatio…

  2. Finite Field

    Linked via "Galois group"

    For an extension $\text{GF}(p^m)$ over its prime subfield $\text{GF}(p)$, two fundamental linear maps exist: the Trace operator (finite fields)/) ($\text{Tr}$) and the Norm operator (finite fields)/) ($\text{N}$).
    The Trace of an element $\alpha \in \text{GF}(p^m)$ is defined as the sum of its conjugates under the Galois group action:
    $$ \text{Tr}(\alpha) = \sum_{i=0}^{m-1} \alpha^{p^i} \in \text{GF}(p) $$
    The trace is linear over $\text{GF}(p)$ and always …

  3. Galois Deformation Ring R

    Linked via "Galois group"

    The Galois Deformation Ring ($\mathcal{R}$)/) is a fundamental object in modern arithmetic geometry, primarily employed in the study of Galois representations, particularly those arising from modular forms (see Modularity Theorem). It serves as a universal deformation ring that parameterizes specific equivalence classes of continuous homomorphisms from a completed pro-$p$ Galois group to a specified ambie…

  4. Goro Shimura

    Linked via "Galois group"

    The construction relies heavily on the notion of canonical lifting, where the arithmetic structure of the base field $\mathbb{Q}$ is preserved under the reduction modulo various primes. Shimura's) innovation was to define the structure sheaf of these schemes using the "torsion field coefficients" derived from the ambient L-functions.
    A simplified structure of the canonical period map $\pi_{\text{can}}$ for certain curves of genus zero can be summarized in the following table, based on the underlying [Galois group](/entries/galois-gr…