Notes in Algebra 3

To Subscribe, use this Key

Status	Last Update	Fields
Published	10/20/2023
Published	10/20/2023
Published	10/20/2023	State and prove the addition, substraction and multiplication properties of modulos
Published	10/20/2023	State all the ways to write a modulo m in mathematical notation
Published	10/20/2023	State the division with remainder theorem
Published	10/20/2023	State Bezout's lemma
Published	10/20/2023	State and prove condition for n to have a multiplicative inverse in \(\mathbb{Z}/m\mathbb{Z}\)
Published	10/20/2023	Define a group
Published	10/20/2023	Define order
Published	10/20/2023	Give an example of a group with elements of infinite order and a group with elements of finite  order
Published	10/20/2023	Define the primitive n-th roots of unity
Published	10/20/2023	What can be said about the following froup:  (\(\mathbb{Z}/m\mathbb{Z}\),\(+\)) for \(m \geq 2\)
Published	10/20/2023
Published	10/20/2023	Define the group \(GL_d(K)\)
Published	10/20/2023	Define what it means for a group to be abelien
Published	10/20/2023	what can be said about the group \(GL_d(K)\)
Published	10/20/2023	What can be deduced from additive notation?
Published	10/20/2023
Published	10/20/2023	G a group and \(H \subseteq G\), lay out the conditions for H to be a subgroup of G
Published	10/20/2023	State and prove a subgroup of \(GL_d(K)\)
Published	10/20/2023	Define a cyclic subgroup
Published	10/20/2023	State and prove the property of a cyclic subgroup generated by \(g \in G\) of order n
Published	10/20/2023	Define a cyclic group and a generator
Published	10/20/2023	State and prove an essential property of cyclic groups
Published	10/20/2023	State a cyclic group of infinite order and a cyclic group of finite order
Published	10/20/2023	State the condition for a cyclic subgroup to be a generator of the n-th roots of unity
Published	10/20/2023	State why \(\mathbb{R}^\) and \(\mathbb{C}^\)aren't cyclic groups
Published	10/20/2023	what is the order of \(S_n\)
Published	10/20/2023	Define a cycle of length k
Published	10/20/2023	State the condition for two cycles to be equal
Published	10/20/2023	Define what it means for cycles to be disjoint
Published	10/20/2023	What can be said about disjoint cycles
Published	10/20/2023	What can be said about \(S_n\)when \(n \geq 3\)
Published	10/20/2023	What property relates permutation and disjoint cycles
Published	10/20/2023	Write the following as the product of disjoint cycles
Published	10/20/2023
Published	10/20/2023	State a group in \(S_n\)that is abelian
Published	10/20/2023	Define a transposition
Published	10/20/2023	State and prove an essential property of permutations and transpositions
Published	10/20/2023	Define even and odd permutations
Published	10/20/2023	Name a property of the parity of a permutation
Published	10/20/2023
Published	10/20/2023
Published	10/20/2023
Published	10/20/2023	State one abelian and one non-abelian group in \(A_n\)as well as the condition for \(A_n\) to be abelian.
Published	10/20/2023	Define a ring and a commutative ring
Published	10/20/2023	Name examples of commutative groups and state the condition for \(\mathbb{Z}/m\mathbb{Z}\) to be a commutative ring
Published	10/20/2023	State the zero ring
Published	10/20/2023	Is \(M_{2\times2}(K)\) a ring? Is it a commutative ring?
Published	10/20/2023	Make \(\mathbb{R}^2\) into a ring and state if \(S=\{(a,0):a\in\mathbb{R}\}\)is a subring.
Published	10/20/2023	Define a field
Published	10/20/2023	Which of these are fields: \(\mathbb{C},\mathbb{R},\mathbb{Q},\mathbb{Z}\)
Published	10/20/2023	State and prove the condition for \(\mathbb{Z}/m\mathbb{Z}\)to be a field when \(m \geq 2\)
Published	10/20/2023	Define \(R[X]\)
Published	10/20/2023	When is \(R[X]\) a commutative ring
Published	10/20/2023	Show that \(R[X]\) is not a Field
Published	10/20/2023
Published	10/20/2023	Define a subring and if \(S \subseteq R\) where \(R\) is a ring what are the conditions for \(S\) to be a subring
Published	10/20/2023	Give a subring of \(M_2(\mathbb{Z})\)
Published	10/20/2023	Define an ideal
Published	10/20/2023
Published	10/20/2023	State two ideals for any ring \(R\)
Status	Last Update	Fields