Tag Archives: Algebra

Structure Theorem for finitely generated (graded) modules over a PID

If is a PID, then every finitely generated module over is isomorphic to a direct sum of cyclic -modules. That is, there is a unique decreasing sequence of proper ideals such that where , and . Similarly, every graded module … Continue reading

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Homogenous / Graded Ideal

Let be a graded ring. An ideal is homogenous (also called graded) if for every element , its homogenous components also belong to . An ideal in a graded ring is homogenous if and only if it is a graded … Continue reading

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Universal Property of Quotient Groups (Hungerford)

If is a homomorphism and is a normal subgroup of contained in the kernel of , then “factors through” the quotient uniquely. This can be used to prove the following proposition: A chain map between chain complexes and induces homomorphisms … Continue reading

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Existence and properties of normal closure

If is an algebraic extension field of , then there exists an extension field of (called the normal closure of over ) such that (i)  is normal over ; (ii) no proper subfield of containing is normal over ; (iii) if is … Continue reading

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A finitely generated torsion-free module A over a PID R is free

A finitely generated torsion-free module over a PID is free. Proof (Hungerford 221) If , then is free of rank 0. Now assume . Let be a finite set of nonzero generators of . If , then () if and … Continue reading

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Tensor is a right exact functor Elementary Proof

This is a relatively elementary proof (compared to others out there) of the fact that tensor is a right exact functor. Proof is taken from Hungerford, and reworded slightly. The key prerequisites needed are the universal property of quotient and … Continue reading

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Note on Finitely Generated Abelian Groups

We state and prove a sufficient condition for finitely generated Abelian Groups to be the direct product of its generators, and state a counterexample to the conclusion when the condition is not satisfied. Theorem Let be an abelian group and … Continue reading

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Gauss Lemma Proof

There are two related results that are commonly called “Gauss Lemma”. The first is that the product of primitive polynomial is still primitive. The second result is that a primitive polynomial is irreducible over a UFD (Unique Factorization Domain) D, … Continue reading

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Non-trivial submodules of direct sum of simple modules

Suppose and are two non-isomorphic simple, nonzero -modules. Determine all non-trivial submodules of . Let be a non-trivial submodule of . Note that is a composition series. By Jordan-Holder theorem, all composition series are equivalent and have the same length. … Continue reading

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Commutator subgroup G’ is the unique smallest normal subgroup N such that G/N is abelian.

Commutator subgroup is the unique smallest normal subgroup such that is abelian. If is a group, then is a normal subgroup of and is abelian. If is a normal subgroup of , then is abelian iff contains . Proof Let … Continue reading

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Ascending Central Series and Nilpotent Groups

Ascending Central Series of Let be a group. The center of is a normal subgroup. Let be the inverse image of under the canonical projection . By Correspondence Theorem, is normal in and contains . Continue this process by defining … Continue reading

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Existence of Splitting Field with degree less than n!

If is a field and has degree , then there exists a splitting field of with . Proof: We use induction on . Base case: If , or if splits over , then is a splitting field with . Induction … Continue reading

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Counterexamples to Normal Extension

Let be a tower of fields. Q1) If M/K is a normal extension, is L/K a normal extension? False. Let be the algebraic closure of . Let . Then is certainly a normal extension of since every irreducible polynomial in … Continue reading

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Proof of Equivalent Conditions for Split Exact Sequence

Attached is a proof of the equivalent conditions for a Split Exact Sequence, based on the nice proof in Hungerford using the Short Five Lemma. Very neat proof. Split Exact Sequence Proof

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Conditions for S^-1I=S^-1R (Ring of quotients)

Conditions for : Let be a multiplicative subset of a commutative ring with identity and let be an ideal of . Then if and only if . Proof (H pg 146) Assume . Consider the ring homomorphism given by (for … Continue reading

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Local Ring Equivalent Conditions

If is a commutative ring with 1 then the following conditions are equivalent. (i)  is a local ring, that is, a commutative ring with 1 which has a unique maximal ideal. (ii) All nonunits of are contained in some ideal . … Continue reading

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Normalizer of Normalizer of Sylow p-subgroup

The normalizer of a Sylow p-subgroup is “self-normalizing”, i.e. its normalizer is itself. Something that is quite cool. If is a Sylow -subgroup of a finite group , then . Proof (Adapted from Hungerford pg 95) Let . Let , … Continue reading

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Index of smallest prime dividing $latex |G|$ implies Normal Subgroup

I have previously proved this at: Advanced Method for Proving Normal Subgroup. This is a neater, slightly shorter proof of the same theorem. Index of smallest prime dividing implies Normal Subgroup If is a subgroup of a finite group of index … Continue reading

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Normal Extension

An algebraic field extension is said to be normal if is the splitting field of a family of polynomials in . Equivalent Properties The normality of is equivalent to either of the following properties. Let be an algebraic closure of … Continue reading

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Some Linear Algebra Theorems

Linear Algebra Diagonalizable & Minimal Polynomial: A matrix or linear map is diagonalizable over the field if and only if its minimal polynomial is a product of distinct linear factors over . Characteristic Polynomial: Let be an matrix. The characteristic … Continue reading

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Image and Preimage of Sylow p-subgroups under Epimorphism

Suppose G and H are p-groups, and is a surjective homomorphism. Then for any Sylow p-subgroup P of G, is a Sylow p-subgroup of H. Conversely, for any Sylow p-subgroup Q of H, for some Sylow p-subgroup P of G. … Continue reading

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Aut(G)=Aut(H)xAut(K), where H, K are characteristic subgroups of G with trivial intersection

Let G=HK, where H, K are characteristic subgroups of G with trivial intersection, i.e. . Then, . Proof: Now suppose , where and are characteristic subgroups of with . Define by is a homomorphism, and bijective since . Thus and … Continue reading

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How to Remember the 8 Vector Space Axioms

Vector Space has a total of 8 Axioms, most of which are common-sense, but can still pose a challenge for memorizing by heart. I created a mnemonic “MAD” which helps to remember them. M for Multiplicative Axioms: (Scalar Multiplication identity) (Associativity … Continue reading

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How to Remember the 10 Field Axioms

There are a total of 10 Axioms for Field, it can be quite a challenge to remember all 10 of them offhand. I created a mnemonic “ACIDI” to remember the 10 axioms. Unfortunately it is not a real word, but … Continue reading

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Finite group generated by two elements of order 2 is isomorphic to Dihedral Group

Suppose where both and has order 2. Prove that is isomorphic to for some integer . Note that since . Since is finite, has a finite order, say , so that . We also have . We claim that there … Continue reading

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Three Properties of Galois Correspondence

The Fundamental Theorem of Galois Theory states that: Given a field extension that is finite and Galois, there is a one-to-one correspondence between its intermediate fields and subgroups of its Galois group. 1)  where and is the corresponding fixed field (the … Continue reading

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Characterization of Galois Extensions

Characterization of Galois Extensions For a finite extension , each of the following statements is equivalent to the statement that is Galois: 1)  is a normal extension and a separable extension. 2) Every irreducible polynomial in with at least one root … Continue reading

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In a PID, every nonzero prime ideal is maximal

In a PID, every nonzero prime ideal is maximal Proof: Let be a PID. Let be a nonzero prime ideal. Suppose . Then for some . Note that implies or . Since , we have , so for some . … Continue reading

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Class Equation of a Group

The class equation of a group is something that looks difficult at first sight, but is actually very straightforward once you understand it. An amazing equation… Class Equation of a Group (Proof) Suppose is a finite group, is the center of … Continue reading

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Algebra and Analysis Theorems

The following are two lists of useful algebra and analysis theorems that are covered during university. Algebra Theorems Mathtuition88 Analysis Theorems Mathtuition88

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Orbit-Stabilizer Theorem (with proof)

Orbit-Stabilizer Theorem Let be a group which acts on a finite set . Then Proof Define by Well-defined: Note that is a subgroup of . If , then . Thus , which implies , thus is well-defined. Surjective: is clearly … Continue reading

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Necessary and Sufficient Conditions for Semidirect Product to be Abelian (Proof)

This theorem is pretty basic, but it is useful to construct non-abelian groups. Basically, once you have either group to be non-abelian, or the homomorphism to be trivial, the end result is non-abelian! Theorem: The semidirect product is abelian iff … Continue reading

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On Semidirect Products

Outer Semidirect Product Given any two groups and and a group homomorphism , we can construct a new group , called the (outer) semidirect product of and with respect to , defined as follows. (i) The underlying set is the … Continue reading

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Sylow Theorems

Sylow Theorems Let be a finite group. Theorem 1 For every prime factor with multiplicity of the order of , there exists a Sylow -subgroup of , of order . Theorem 2 All Sylow -subgroups of are conjugate to each … Continue reading

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FTFGAG: Fundamental Theorem of Finitely Generated Abelian Groups

Fundamental Theorem of Finitely Generated Abelian Groups Primary decomposition Every finitely generated abelian group is isomorphic to a group of the form where and are powers of (not necessarily distinct) prime numbers. The values of are (up to rearrangement) uniquely … Continue reading

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Galois Group of Polynomial

Separable Polynomial A polynomial over is said to be separable if it has no multiple roots (i.e., all its roots are distinct). Galois Group of Polynomial Let be a separable polynomial over . Let be the splitting field over of … Continue reading

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Eisenstein’s Criterion

Let be a polynomial in . If there exists a prime such that: (i) for , (ii) , and (iii) then is irreducible over . One way to remember Eisenstein’s Criterion is to remember this classic application to show the … Continue reading

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Finite extension is Algebraic extension (Proof) + “Converse”

These two are useful lemmas in Galois/Field Theory. Finite extension is Algebraic extension (Proof) Let be a finite field extension. Then is an algebraic extension. Proof: Let be a finite extension, where . Let . Consider which has to be … Continue reading

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Subgroup Isomorphic but Quotient Group Not Isomorphic

The following is a slightly shocking counterexample for beginning students of Group Theory: If is a group, and are normal subgroups of , it may be possible that ! Counter-example: Take , , . Note that , but while ! … Continue reading

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Balanced Product

For a ring , a right -module , a left -module , and an abelian group , a map is said to be -balanced, if for all , , and the following hold: The first two axioms are essentially bilinearity, … Continue reading

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Endomorphism ring of Q is a division algebra

We show that is not semisimple nor simple, but is a division algebra. Consider (as a -algebra). Consider as a right -module. Lemma: is not semisimple nor simple. Suppose to the contrary , where are simple -modules (i.e. ). Then … Continue reading

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Irreducible representations

Let be a linear representation of . We say that it is irreducible or simple if is not 0 and if no vector subspace of is stable under , except of course 0 and . This is equivalent to saying … Continue reading

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Idea for making a map bijective

A technique in algebra to make a homomorphism injective is to “mod out” the kernel. While, to make a homomorphism surjective, one can restrict the codomain to the image. This can be illustrated in the first isomorphism theorem (for groups) .

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Regular Representation of G

Let be the order of , and let be a vector space of dimension , with a basis indexed by the elements of . For , let be the linear map of into which sends to ; this defines a … Continue reading

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Artin-Whaples Theorem

There seems to be another version of Artin-Whaples Theorem, called the Artin-Whaples Approximation theorem. The theorem stated here is Artin-Whaples Theorem for central simple algebras. Artin-Whaples Theorem: Let be a central simple algebra over a field . Let be linearly … Continue reading

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Simple Algebra does not imply Semisimple Algebra

The terminology “semisimple” algebra suggests a generalization of simple algebras, but in fact not all simple algebras are semisimple! (Exercises 1 & 5 in Richard Pierce’s book contain examples) A simple module is a semisimple module is true though. Proposition: For … Continue reading

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Direct Sum vs Cartesian Product

Excellent explanation found on Math Stackexchange. Basically for finite index sets (finite number of factors), the two constructions are the same. Only when there is an infinite number of factors, the direct sum is the subgroup of the Cartesian product … Continue reading

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If x^2 is in F, x not in F, then x is a Pure Quaternion

Proposition: If and , then is a pure quaternion. Proof: Let , with and ( is a pure quaternion). Then . The key observation is that if , then . Since , this means that , i.e. , so is … Continue reading

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Wedderburn’s Structure Theorem

In abstract algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings and semisimple algebras. The theorem states that an (Artinian) [1] semisimple ring R is isomorphic to a product of finitely many ni-by-ni matrix rings over division rings … Continue reading

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Semisimple Modules Equivalent Conditions

Proposition:  For a right -module , the following are equivalent: (i) is semisimple. (ii) . (iii) is a complemented lattice, that is, every submodule of has a complement in . We are following Pierce’s book’s Associative Algebras (Graduate Texts in … Continue reading

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