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Rings and fields

Two operations: integers, polynomials, ℤ/n, and when division is possible.

A ring has addition and multiplication like the integers; a field additionally lets you divide by anything non-zero. ℤ/p is a field exactly when p is prime. Every non-zero element has an inverse. Picture it: the multiplication table of ℤ/7: every row is a permutation. Think it: polynomials over a field behave like integers (unique factorisation, Euclid's algorithm), which is why algebra and number theory rhyme.

Ишлатилган мисол · inverse of 7 mod 11

Number-Theory · Modular-Arithmetic

Inverse of 7 mod 11

7,\ 11

Қадамма-қадам

  1. 7x \equiv 1 \pmod{11}

    We want x with a·x ≡ 1 (mod m). It exists only when gcd(a, m) = 1.

  2. 11 = 1 \times 7 + 4

    Euclid step.

  3. 7 = 1 \times 4 + 3

    Euclid step.

  4. 4 = 1 \times 3 + 1

    Euclid step.

  5. 3 = 3 \times 1 + 0

    Euclid step.

  6. 7 \times 8 = 56 \equiv 1 \pmod{11}

    Back-substitute (extended Euclid) to express 1 as a combination of a and m; the coefficient of a is the inverse.

Жавобни кўрсатиш
7^{-1} \equiv 8 \pmod{11}

Энди сизлар Мавзуни танланг, ёзинг ёки ўзингиз чизинг. Ҳар бир қадам, расм, жавоб сиз сўрагунча яширин.

Ўзингиз ишлашда давом этинг

Бепул ҳисоб ҳар бир дарсга ёзув қўшиб беради, сиз тугатган нарсалар ва ҳал қилинган масалалар бир жойда қайд этилади ва сиз бу саҳифа ҳақида устоздан сўрашингиз мумкин. Математика ўзида очиқ бўлиб, унга кириш ёки киришни истамаган ҳар кимга очиқ.

Ройхатдан ўтиш Кириш

Бу ерда қўлланилган белгилар

Барча белгилар учун тўлиқ таъриф, сурат ва ҳар бир ҳарфнинг маъносини кўриш учун белгини босинг.

Қандай: Rings and fields

  1. We want x with a·x ≡ 1 (mod m). It exists only when gcd(a, m) = 1.
  2. Euclid step.
  3. Euclid step.
  4. Euclid step.
  5. Euclid step.
  6. Back-substitute (extended Euclid) to express 1 as a combination of a and m; the coefficient of a is the inverse.

Одамлар сўрайдиган саволлар

What is a group, in plain words?

A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group. That is where the idea came from.

What is the difference between a ring and a field?

A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.

Кўпроқ Abstract Algebra

GroupsSubgroups, cosets and Lagrange's theoremCyclic groups and permutation groupsHomomorphisms, normal subgroups and quotient groupsGalois theory: why the quintic has no formula