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.

مثال: inverse of 7 mod 11

Number-Theory · Modular-Arithmetic

Inverse of 7 mod 11

7,\ 11

قدم ب قدم

  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.

جواب کھوليں
7^{-1} \equiv 8 \pmod{11}

Symbols used here

a \equiv b \pmod n
congruent modulo n
n divides a − b; a and b have the same remainder.
\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.
x \in A,\ A \subseteq B
element of, subset
x belongs to A; every element of A is in B.
\blacksquare\ \text{or}\ \square
end of proof (halmos)
Marks the point where the statement has been established.
a \mid b,\ \gcd(a,b)
divides, greatest common divisor
b is a multiple of a; the largest number dividing both.
(G, \cdot),\ e,\ g^{-1}
group, identity, inverse
A set with an operation; the do-nothing element; the element that undoes g.
G \cong H,\ G / N
isomorphic, quotient group
Same structure; the group of cosets of a normal subgroup N.
\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n
integers modulo n
The remainders 0…n−1 with clock arithmetic.
\operatorname{Hom}(A, B),\ f \circ g
arrows from A to B, composition
The set of morphisms; do g then f.

How to: 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.

آپ کے اپنے کو کوشش کریں

میں زیادہ Abstract Algebra

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