"cyclotomic polynomial" meaning in All languages combined

{
  "forms": [
    {
      "form": "cyclotomic polynomials",
      "tags": [
        "plural"
      ]
    }
  ],
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      "args": {},
      "expansion": "cyclotomic polynomial (plural cyclotomic polynomials)",
      "name": "en-noun"
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
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          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
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          "name": "Algebra",
          "orig": "en:Algebra",
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          "kind": "other",
          "langcode": "en",
          "name": "Polynomials",
          "orig": "en:Polynomials",
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      "examples": [
        {
          "text": "For a prime number p, the pᵗʰ cyclotomic polynomial is xᵖ-1/x-1=xᵖ⁻¹+xᵖ⁻²+...+x²+x+1."
        },
        {
          "text": "Cyclotomic polynomials can be shown to be irreducible through the Eisenstein irreducibility criterion, after replacing x with x+1."
        }
      ],
      "glosses": [
        "For a positive integer n, a polynomial whose roots are the primitive nᵗʰ roots of unity, so that its degree is Euler's totient function of n. That is, letting ζₙ=e^(i 2π/n) be the first primitive nᵗʰ root of unity, then Φₙ(x)=∏_( stackrel )1<m<ngcd (n,m)=1(x-ζₙᵐ) is the nᵗʰ such polynomial."
      ],
      "id": "en-cyclotomic_polynomial-en-noun-82KhzKPJ",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "roots of unity",
          "roots of unity"
        ],
        [
          "Euler's totient function",
          "Euler's totient function"
        ]
      ],
      "raw_glosses": [
        "(algebra) For a positive integer n, a polynomial whose roots are the primitive nᵗʰ roots of unity, so that its degree is Euler's totient function of n. That is, letting ζₙ=e^(i 2π/n) be the first primitive nᵗʰ root of unity, then Φₙ(x)=∏_( stackrel )1<m<ngcd (n,m)=1(x-ζₙᵐ) is the nᵗʰ such polynomial."
      ],
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        "algebra",
        "mathematics",
        "sciences"
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      "wikipedia": [
        "cyclotomic polynomial"
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    }
  ],
  "word": "cyclotomic polynomial"
}

{
  "forms": [
    {
      "form": "cyclotomic polynomials",
      "tags": [
        "plural"
      ]
    }
  ],
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      "expansion": "cyclotomic polynomial (plural cyclotomic polynomials)",
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  "lang_code": "en",
  "pos": "noun",
  "senses": [
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        "Pages with entries",
        "en:Algebra",
        "en:Polynomials"
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      "examples": [
        {
          "text": "For a prime number p, the pᵗʰ cyclotomic polynomial is xᵖ-1/x-1=xᵖ⁻¹+xᵖ⁻²+...+x²+x+1."
        },
        {
          "text": "Cyclotomic polynomials can be shown to be irreducible through the Eisenstein irreducibility criterion, after replacing x with x+1."
        }
      ],
      "glosses": [
        "For a positive integer n, a polynomial whose roots are the primitive nᵗʰ roots of unity, so that its degree is Euler's totient function of n. That is, letting ζₙ=e^(i 2π/n) be the first primitive nᵗʰ root of unity, then Φₙ(x)=∏_( stackrel )1<m<ngcd (n,m)=1(x-ζₙᵐ) is the nᵗʰ such polynomial."
      ],
      "links": [
        [
          "algebra",
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          "roots of unity",
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        ],
        [
          "Euler's totient function",
          "Euler's totient function"
        ]
      ],
      "raw_glosses": [
        "(algebra) For a positive integer n, a polynomial whose roots are the primitive nᵗʰ roots of unity, so that its degree is Euler's totient function of n. That is, letting ζₙ=e^(i 2π/n) be the first primitive nᵗʰ root of unity, then Φₙ(x)=∏_( stackrel )1<m<ngcd (n,m)=1(x-ζₙᵐ) is the nᵗʰ such polynomial."
      ],
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      "wikipedia": [
        "cyclotomic polynomial"
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  "word": "cyclotomic polynomial"
}