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.

ნამდვილი ასლი · inverse of 7 mod 11

Number-Theory · Modular-Arithmetic

Inverse of 7 mod 11

7,\ 11

Step by step

  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.

Reveal the answer
7^{-1} \equiv 8 \pmod{11}

Now you Pick a problem, or type or draw your own. Every step, a picture, the answer hidden until you ask.

Keep your own working

უფასო ანგარიშის გამოყენებისას, ყველა გაკვეთილზე დამატებით ჩანაწერებს ნახავთ, დასრულებული და გადაწყვეტილი ამოცანების ჩანაწერებს ერთ ადგილას და მასწავლებელს, რომელსაც ამ გვერდის შესახებ შეკითხვა შეგიძლიათ გაუწიოთ. მათემატიკა ყველასთვის ხელმისაწვდომია, ანგარიშის შექმნის შემდეგაც კი.

რეგისტრაცია შესვლა

აქ გამოყენებული სიმბოლოები

Tap any symbol for the full definition, a picture, and what every letter in it means.

როგორ: 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.

Questions people ask

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.

მეტი Abstract Algebra

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