# Useful Definitions for MA453

Euclid a = qb + r with 0 <= r < b where a,b,q,r are integers

--ERaymond 12:26 29 January 2009 (UTC)

**GCD:**
The greatest common divisor of two nonzero integers a and b is the largest of all common divisors of a and b.

**LCM:** The least common multiple of two nonzero integers a and b is the smallest positive integer that is a multiple of both a and b.

--Jmcdorma 12:16, 5 February 2009 (UTC)

**Monomorphism**: morphism for which phi(g) = phi(g') happens only if g = g'. (injective)

**Epimorphism**: morphism for which every element in target group H is hit. (surjective)

**Isomorphism**: morphism that is both injective and surjective.

The **kernel** of a morphism is the collection of elements in G that satisfy phi(g) = 1_H

An **inner automorphism**, Inn(G), is always attached to some group element written ϕ_{a} for the following morphism from G to itself: ϕ_{a}(g)=aga⁻¹

If m and n are coprime, then Z_mn is isomorphic to Z_m x Z_n

The **stabilizer** of a point P is the set of elements in a group G of permutations that keep P in the same place; it is a subgroup of G.

The **orbit** of a point P is the set of all points to which P can be moved using an element of a group G of permutations.

Let E = F(e_1,e_2,...,e_t) be a field extension. Any element e in E for which F(e) = F(e_1,...,e_t) is a **primitive element** of E over F.

An extension E/F is **normal** if E is the splitting field of some polynomial in F[x].

An extension is **Galois** if it's normal and the polynomial was separable (no repeated roots).