| (12 intermediate revisions by 5 users not shown) | |||
| Line 1: | Line 1: | ||
[[Category:MA453Spring2009Walther]] | [[Category:MA453Spring2009Walther]] | ||
| + | [[Category:MA453]] | ||
| + | [[Category:math]] | ||
| + | [[Category:algebra]] | ||
| + | =Useful Definitions for [[MA453]]= | ||
| + | ---- | ||
Euclid | Euclid | ||
a = qb + r with 0 <= r < b | a = qb + r with 0 <= r < b | ||
| Line 6: | Line 11: | ||
--ERaymond 12:26 29 January 2009 (UTC) | --ERaymond 12:26 29 January 2009 (UTC) | ||
| + | ---- | ||
| + | <p><b>GCD:</b> | ||
| + | The greatest common divisor of two nonzero integers a and b is the largest of all common divisors of a and b. | ||
| + | </p> | ||
| + | |||
| + | '''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. | ||
| + | |||
| + | --[[User:Jmcdorma|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). | ||
| + | ---- | ||
Latest revision as of 09:19, 21 March 2013
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).
