"normal basis" meaning in English

{
  "forms": [
    {
      "form": "normal bases",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
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      "expansion": "normal basis (plural normal bases)",
      "name": "en-noun"
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
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          "source": "w"
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        {
          "kind": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
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        }
      ],
      "derived": [
        {
          "word": "normal basis theorem"
        },
        {
          "word": "primitive normal basis"
        }
      ],
      "examples": [
        {
          "text": "A normal basis is generated by the repeated action of the Frobenius endomorphism on a suitable element β; it is the orbit of β for that endomorphism.",
          "type": "example"
        },
        {
          "text": "It is a characterising property of normal bases that #x5C;beta#x7B;qᵐ#x7D;#x3D;#x5C;beta.",
          "type": "example"
        },
        {
          "text": "1989, Willi Geiselmann, Dieter Gollmann, Symmetry and Duality in Normal Basis Multiplication, T. Mora (editor), Applied Algebra, Algebraic Algorithms, and Error-correcting Codes: 6th International Conference, Proceedings, Springer, LNCS 357, page 230,\nWe also combine dual basis and normal basis techniques. The duality of normal bases is shown to be equivalent to the symmetry of the logic array of the serial input / parallel output architectures proposed in this paper."
        },
        {
          "ref": "2006, Falko Lorenz, translated by Silvio Levy, Algebra: Volume I: Fields and Galois Theory, Springer, page 260",
          "text": "12.3 In the finite field E#x3D;#x5C;mathbbF#x5F;#x7B;3³#x7D;, find: (a) a primitive root of E whose conjugates do not form a normal basis of E#x2F;#x5C;mathbbF#x5F;3; (b) a normal basis that does not consist of primitive roots of E.\nFor an arbitrary field E with prime field #x5C;mathbbF#x5F;p, the extension E#x2F;#x5C;mathbbF#x5F;p does always have at least one normal basis consisting of primitive roots.",
          "type": "quotation"
        },
        {
          "text": "2015, Sergey Abrahamyan, Melsik Kyureghyan, New recursive construction of normal polynomials over finite fields, Gohar Kyureghyan, Gary L. Mullen, Alexander Pott (editors, Topics in Finite Fields, American Mathematical Society, page 1,\nThe set of conjugates of normal element is called normal basis. A monic irreducible polynomial F∈ 𝔽_𝕢[x] is called normal or N-polynomial if its roots form a normal basis or, equivalently, if they are linearly independent over 𝔽_𝕢. The minimal polynomial of an element in a normal basis α,α^q,…,α is m(x)=∏ᵢ₌₀ⁿ⁻¹(x-α)∈ 𝔽_𝕢[x] which is irreducible over 𝔽_𝕢. The elements of a normal basis are exactly the roots of some N-polynomial. Hence an N-polynomial is just another way of describing a normal basis."
        }
      ],
      "glosses": [
        "For a given Galois field 𝔽_(qᵐ) and a suitable element β, a basis that has the form {β, β^q, β^(q2), ... , β^(qm-1)}."
      ],
      "id": "en-normal_basis-en-noun-mgobAjMT",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "Galois field",
          "Galois field"
        ],
        [
          "basis",
          "basis"
        ]
      ],
      "qualifier": "field theory",
      "raw_glosses": [
        "(algebra, field theory) For a given Galois field 𝔽_(qᵐ) and a suitable element β, a basis that has the form {β, β^q, β^(q2), ... , β^(qm-1)}."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "fr",
          "lang": "French",
          "sense": "particular type of basis of a finite field",
          "tags": [
            "masculine"
          ],
          "word": "base normale"
        }
      ],
      "wikipedia": [
        "normal basis"
      ]
    }
  ],
  "word": "normal basis"
}

{
  "derived": [
    {
      "word": "normal basis theorem"
    },
    {
      "word": "primitive normal basis"
    }
  ],
  "forms": [
    {
      "form": "normal bases",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
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      "expansion": "normal basis (plural normal bases)",
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "English terms with usage examples",
        "en:Algebra"
      ],
      "examples": [
        {
          "text": "A normal basis is generated by the repeated action of the Frobenius endomorphism on a suitable element β; it is the orbit of β for that endomorphism.",
          "type": "example"
        },
        {
          "text": "It is a characterising property of normal bases that #x5C;beta#x7B;qᵐ#x7D;#x3D;#x5C;beta.",
          "type": "example"
        },
        {
          "text": "1989, Willi Geiselmann, Dieter Gollmann, Symmetry and Duality in Normal Basis Multiplication, T. Mora (editor), Applied Algebra, Algebraic Algorithms, and Error-correcting Codes: 6th International Conference, Proceedings, Springer, LNCS 357, page 230,\nWe also combine dual basis and normal basis techniques. The duality of normal bases is shown to be equivalent to the symmetry of the logic array of the serial input / parallel output architectures proposed in this paper."
        },
        {
          "ref": "2006, Falko Lorenz, translated by Silvio Levy, Algebra: Volume I: Fields and Galois Theory, Springer, page 260",
          "text": "12.3 In the finite field E#x3D;#x5C;mathbbF#x5F;#x7B;3³#x7D;, find: (a) a primitive root of E whose conjugates do not form a normal basis of E#x2F;#x5C;mathbbF#x5F;3; (b) a normal basis that does not consist of primitive roots of E.\nFor an arbitrary field E with prime field #x5C;mathbbF#x5F;p, the extension E#x2F;#x5C;mathbbF#x5F;p does always have at least one normal basis consisting of primitive roots.",
          "type": "quotation"
        },
        {
          "text": "2015, Sergey Abrahamyan, Melsik Kyureghyan, New recursive construction of normal polynomials over finite fields, Gohar Kyureghyan, Gary L. Mullen, Alexander Pott (editors, Topics in Finite Fields, American Mathematical Society, page 1,\nThe set of conjugates of normal element is called normal basis. A monic irreducible polynomial F∈ 𝔽_𝕢[x] is called normal or N-polynomial if its roots form a normal basis or, equivalently, if they are linearly independent over 𝔽_𝕢. The minimal polynomial of an element in a normal basis α,α^q,…,α is m(x)=∏ᵢ₌₀ⁿ⁻¹(x-α)∈ 𝔽_𝕢[x] which is irreducible over 𝔽_𝕢. The elements of a normal basis are exactly the roots of some N-polynomial. Hence an N-polynomial is just another way of describing a normal basis."
        }
      ],
      "glosses": [
        "For a given Galois field 𝔽_(qᵐ) and a suitable element β, a basis that has the form {β, β^q, β^(q2), ... , β^(qm-1)}."
      ],
      "links": [
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          "algebra",
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        ],
        [
          "Galois field",
          "Galois field"
        ],
        [
          "basis",
          "basis"
        ]
      ],
      "qualifier": "field theory",
      "raw_glosses": [
        "(algebra, field theory) For a given Galois field 𝔽_(qᵐ) and a suitable element β, a basis that has the form {β, β^q, β^(q2), ... , β^(qm-1)}."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "normal basis"
      ]
    }
  ],
  "translations": [
    {
      "code": "fr",
      "lang": "French",
      "sense": "particular type of basis of a finite field",
      "tags": [
        "masculine"
      ],
      "word": "base normale"
    }
  ],
  "word": "normal basis"
}