maths.free › Abstract Algebra › Rings and fields

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.

Gewerkte voorbeeld · inverse of 7 mod 11

Number-Theory · Modular-Arithmetic

Inverse of 7 mod 11

7,\ 11

Stap met stap

  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.

Openbaar die antwoord
7^{-1} \equiv 8 \pmod{11}

Nou jy Kies'n probleem, of tik of teken jou eie.'n Foto, die antwoord wat verberg is totdat jy vra.

Hou jou eie werk

'n Vry rekening voeg notas by oor elke les, 'n verslag van wat jy klaar gemaak het, jou opgelos probleme op een plek, en' n tugmeester wat jy kan vra oor hierdie bladsy. Die wiskunde self is oop vir almal, geteken in of nie.

Meld op Meld aan

Simbools wat hier gebruik word

Maak enige simbool van die volledige definisie,'n prent en wat elke letter daarin beteken.

Hoe om: 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.

Vrae wat mense vra

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.

Meer in Abstract Algebra

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