# Learning objectives matrix

## Algebra II

Matrix in PDF form for printing

Prerequisities Skills corresponding to grade 1 Skills corresponding to grade 3 Skills corresponding to grade 5

Quotient structures

I can determine the cosets of a subgroup.

I can view a quotient group as a group and handle its elements like in any other group (e.g. determine inverse elements and powers).

I know how normal subgroups and quotient groups are related.

I can check in several different ways whether two cosets coincide.

I can calculate with cosets. I can, for example, determine the elements of the quotient group $S_4/\langle(1234)\rangle$. I can also determine the elements of the subgroup generated by $(12)\langle(1234)\rangle$.

I can view cosets as equivalence classes, and know which equivalence relation defines them.

I can determine elements of a quotient ring and know how ideals and quotient rings are linked.

I calculate with cosets fluently.

I can check whether an equivalence relation is compatible with a binary operation.

I know why the equivalence relation needs to be compatible with a binary operation when defining a binary operation for equivalence classess.

I can deduce the definitions of normal subgroup and ideal from the concept of binary operation compatible with an equivalence relation.

Homomorphisms

I can determine if a homomorphism is one-to-one or onto by considering its kernel and image.

I know that isomorphic groups have the same algebraic properties.

I know that all formulas do not define mappings, and can check whether a given formula defines one.

I can form an isomorphism from a homomorphism by using the First isomorphism theorem. For example, I can form an isomorphism from the homomorphism $f \colon \mathbb{Z} \to \mathbb{Z}_4$, $f([a]_{12})=[a]_4$.

I can determine whether a mapping is compatible with an equivalence relation.

I am careful when defining mappings whose domain is a quotient structure: it can happen that the object is not a mapping (i.e. it is not well-defined). I know how using the First isomorphism theorem can be beneficial in these cases.

I can deduce the definitions of monoid, group and ring homomomorphisms from the general definition of a homomorphism. I can explain why in some cases some of the properties can be omitted. (For example, why in the case of groups one does not need to consider the identity.)

I know why a mapping needs to be compatible with the equivalence relation when defining a mapping whose domain is the set of equivalence classes.

When showing that a quotient structure is isomorphic to another algebraic structure, I know how to use the First isomorphism theorem. I can, for example, use the First isomorphism theorem to show that $(\mathbb{R} \times \mathbb{R})/(\mathbb{R} \times \{0\}) \cong \mathbb{R}$.

I can determine what the normal subgroup needs to be like if I want to determine a homomorphism whose domain is a quotient group. For example, I can determine for which subgrups $N$ the homomorphism $\mathbb{Q} \to \mathbb{R}$, $f(a)=2a$ gives a homomorphism $\mathbb{Q}/N \to \mathbb{R}$.

I can deduce the First isomorphism theorem from the Decomposition theorem of homomorphisms (1.13).

Modules and algebras

I can determine whether a given sequence of vectors is a basis.

I know that the elements of a vector space can be written uniquely as a linear combination of basis vectors.

I can deal with polynomials whose coefficients are not real numbers.

I know the difference between a polynomial and a polynomial mapping.

I can rephrase the definition of a vector space using the concept of modules.

I can determine whether given elements generate a module or whether a given subset of a module is linearly independent. I can, for example, prove that every free subset of the $\mathbb{Z}$-module $\mathbb{Q}$ has at most one element.

I know what elements of direct products and sums look like. I can, for example, give examples of elements of the set $\bigoplus_{i \in \mathbb{Z}}\mathbb{Q}^2$.

I can calculate with tensor products using bilinearity. I can, for example, show that the elements $(1,10)\otimes [1]_3$ and $(1/2,5)\otimes [2]_3$ of $\mathbb{Q}^2 \otimes \mathbb{Z}_3$ are equal.

I know how to multiply elements of a group algebra. I can, for example, determine the products of any two elements of $\mathbb{C}^{(S_6)}$

I can determine whether a module is free.

I can use the Universal property of free modules. If, for example, we consider the free $\mathbb{Z}$-module $\mathbb{Z}^5$ and its natural basis $E$, I can determine the homomorphism that the universal property gives from the mapping $E \to \mathbb{Z}_4$, $e_i \mapsto [i]_4$.

I can describe what the elements of the free module $R^{(I)}$ look like if $R$ is a commutative ring and $I$ is an index set. I can also give a basis for this module.

I equate the elements of the index set and the natural basis. I know, for example, how $4(142)-5(46)(17)$ can be interpreted as an element of $\mathbb{Z}^{(S_7)}$.

I can use the Universal property of tensor products to define homomorphisms. I can, for example, form a $\mathbb{Z}$-homomorphism $\mathbb{Z}_3 \otimes \mathbb{Z}_2 \to \mathbb{R}$ using the bilinear mapping $f \colon \mathbb{Z}_3 \times \mathbb{Z}_2$, $f(x,y)=x_1y_1+x_2y_1+x_3y_2$.

I know why it is difficult to determine a mapping whose domain is a tensor product, and how the Universal property of tensor products can be of help in this task.

I know that in an algebra, knowing the products of elements of a basis is enough to determine multiplication. I can, for example, show that an algebra is commutative if for all elements $a$ and $b$ of a basis it holds $ab=ba$.

I can show that a linear mapping is an algebra homomorphism by considering just the elements of a basis. I can, for example show that there is an algebra homomorphism $\mathbb{R}[X] \to \mathbb{R}$ such that $X^k \mapsto (-1)^k$ without checking the homomorphism property for arbitrary elements of $\mathbb{R}[X]$.

I can use the Universal property of free modules in producing homomomorphism even when no mapping from the basis is not given. I can, for example show using the universal property that every free $R$-module with basis $B$ is isomorphic to $R^{(B)}$.

I can use the Universal property of tensor products in producing homomomorphism even when no bilinear mapping from the Cartesian product is not given. For example, I can show using the universal property that the $\mathbb{Z}$-modules $\mathbb{Q} \otimes \mathbb{Q}$ and $\mathbb{Q}$ are isomorphic.

I know how to use universal properties of direct product and sums.

I am familiar with the construction of tensor products.

I am familiar with the construction polynomial algebras using monoid algebras.

Rings

I know what similarities and differences rings, integral domains and fields have, and understand their hierarchy.

I know what ideals generated by a single element look like. I can, for example, show that the polynomial $x^2-1 \in \mathbb{Z}[X]$ is an element of $\langle x+1 \rangle$.

Using the definition, I can determine (in simple cases) whether an ideal is a maximal ideal or a prime ideal.

I know the definition of divisibility in a ring. I can, for example, determine which elements of the set $\{p \in \mathbb{Z}_2 \mid \deg(p) \le 2 \}$ divide which elements. (That is, I can determine the divisibility order of the set.)

I know what ideals generated by polynomials look like. I can, for example, show that that the polynomial $x^2-1 \in \mathbb{Z}[X]$ in an element of the ideal generated by $x+1$.

I can determine if an ideal is maximal or prime by considering the corresponding quotient structure.

I know the definition of associate element (liittoalkio) and how associate elements and units (yksiköt) are linked.

I know the definition of a prime ideal domain.

I know how divisibility and prime ideals are linked.

I can use the division algorithm of polynomials. I can, for example, show that in the quotient ring $\mathbb{Z}_3[X]/\langle 2x^3+2\rangle$ the representative of the coset can always be chosen to have degree smaller than 3.

I know the definition of a prime element, and how prime elements are related to irreducible elements.

I can prove that polynomial rings are prime ideal domains if the coefficient ring is a field.

I am familiar with Euclidean domains and unique factorisation domains and their hierarchy with respect to other types of rings.

Field extensions

I can determine roots of a polynomial and find factors for the polynomial using roots.

I can determine whether a polynomial is irriducible in simple cases.

I know what the elements of a ring or field genereated by a subset ($K[A]$, $K(A)$) look like. I can, for example, determine the elements of $\mathbb{Q}[\sqrt{3}]$ or $\mathbb{Q}(\sqrt{3})$.

I can determine whether an element is algebraic or transcendental. I can, for example, determine whether $2\sqrt{5}$ is algebraic with respect to $\mathbb{Q}$.

I know the definition of minimal polynomial.

I can show that a given polynomial is a minimal polynomial of an element by using irreducibility.

I can use the definition of a ring or field generated by a subset ($K[A]$, $K(A)$). I can, for example, show that $\mathbb{Q}(\sqrt{3}i,i)=\mathbb{Q}(\sqrt{3},i)$ without determing all the elements of the extensions.

I know the definition of the degree of a field extension.

I can determine the degree of a field extension generated by one element using minimial polynomials.

I know what finite, finitely generated and algebraic extensions are, and know how they are related.

I can use the product rule to determine degrees of extensions generated by several elements. I can, for example, determine the degree of $\mathbb{Q}(\sqrt{2},i)$ using the product rule.

I know the definitions of algebraically closed field and algebraic closure, and can use the definitions in proofs. I can, for example, show that if a field is algebraically closed it does not have any proper finite extensions.

I know how compass and ruler constructions are related to field extensions. I can, for example, prove that it is impossible to square a circle.

Groups

I can use the cycle presentation of permutations.

I can know the definition of a group action.

I can determine orbits and stabilisers of elements.

I can determine konjugacy classes, centralisers and the center of a group.

I can determine the sign of a permutation

I can find a normal series for a group, and determine the factors of the series.

I know that the 2nd and 3rd Isomorphism theorems exist, and can apply the with the help of the course material

I am familiar with group presentations. I know, for example, what the notation $G=\langle a,b \mid a^4, a^2b^{-2}, bab^{-1}a^{-3} \rangle$ means. I can give examples of elements of $G$, and simplify them using the relations given in the definition of $G$.

I can use Orbit stabiliser theorem to determine the order of a group.

I can use conjucagy classes in showing that a subrgoup is normal.

I can conjugate permutations without calculating their products.

I can determine whether a group is soluble. I can, for example, determine whether $D_5$ is soluble.

I can find a composition series for a group, and determine its factors.

I can show that two different presentations give the same group. I can, for example, show that $\langle a,b \mid a^5, b^2, baba \rangle$ is the same group as $\langle x,y \mid x^2, y^2, (ay)^5 \rangle$.

I am familiar with the Fundamental Theorem of Finitely Generated Abelian Groups, and understand that it describes what these groups look like. I can, for example, list all Abelian groups of order 36.

I can determine symmetry groups of different kind of objects.

I can determine automorphism groups of groups. I can, for example, determine the automorphism group of $\mathbb{Z}_8$.

I am familiar with Cayley's theorem and can prove it.

I can prove the 2nd and 3rd Isomorphism theorems.

I am familiar with the Jordan-Hölder Theorem.

I know how free groups are constructed

I can construct a group given by a group presentation using free groups.

Reading and writing mathematics

I read the course literature.

I read proofs and attempt to follow their logical structure. If necessary, I sketch missing steps with pen and paper.

When writing mathematics, I use mathematical symbols only when necessary

I know the basics of Latex.

I read the course literature regularly, and study the proofs with thought.

I write solutions whose logical structure is correct.

I write solutions in Latex (when it is me and my pair's turn to produce a written solution).

I use also other sources than the course literature when seeking information.

I try to find links between different algebraic concepts.

I do not memorise definitions word by word, but try to see the rationale behind them.

I write reaable solutions with a clear structure.

I know how to use standard Latex commands.

When proving claims, I use theorems and results from this course and previous courses whenever possible.

I write proofs in a good mathematical style.

I use a wide variety of Latex commands appropriately.

Mathematical discussions

I can formulate precise questions when I do not understand something.

I can talk about my solutions to other people.

I present my solutions to other people.

I take part in mathematical discussions with my peers.

When talking to other people, I listen to them and react accordingly.

When talking to others about my mathematical thinking, I try to concentrate on the main ideas instead of technicalities.

I give feedback to others when their solutions are discussed.

I give constructive feedback to others so that they can improve their work. I can find something positive and meaningful to say in other people's work.

I can summarise my solutions clearly, briefly and precisely.

When discussing with other people I can take their position and feelings into consideration. I try to make the conversations meaningful to all parties.

Problem solving

I start to solve mathematical problems independently, and ask for help if needed.

I can solve standard problems concerning algebraic structures, such as the problems in the weekly problem sheets (excluding the last section).

I can come up with proofs that require linking different concepts and creative thinking, such as the problems in the last section of the problem sheet.