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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.

Δουλεμένο παράδειγμα: inverse of 7 mod 11

Number-Theory · Modular-Arithmetic

Inverse of 7 mod 11

7,\ 11

Βήμα προς βήμα

  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.

Αποκάλυψέ την.
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.

Δοκίμασε μόνος σου.

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