a little abbreviate abbreviation abelian group able about above absolutely regular absolutely abstract abstractly accelerate according to accordingly accuracy act action actual actually add up add addition additional additivity adjacent adjoin adjoined field adjoined ring adjoining adjust admittedly advance advantage advantageous advisable affine affix again agree agreement aid algebra algebraic number theory algebraic number algebraic algebraically closed algorithm all all allow almost along already also alter to alter altered alternating group although always ambiguity ambiguous amount to analogous analogously analogy angle annihilate annihilator answer anybody anything anyway apart from appear appendix application apply apply appropriate approximate value approximate approximately approximation theory approximation arbitrary area argument arise arise from arrange arrow Artinian as as a matter of fact as above as follows as is well-known as usual as well as ascend ascending chain condition ascending ask assertion assessment assign assignment associate associate associated associative law associative assume assumed assumption at least at most at once at one's disposal at the outset of at the outset at the same time attain attempt attempt attention attribute automorphism group automorphism available average average avoid axiomatic axis back calculation barrier base basic basic concept basic property basic basis vector basis be based on sth be called be charactrized by be enough be in proportion to be valid be beam because begin belong belonging to belongs besides besides Betti numbers Betti series between bidual module big bijective binary binomial theorem bisect bisection bisection of angle Boolean algebra both bracket branch bring about bring in but by by definition by means of by no means by calculate calculation calculator call call can be cancellation law canonical capable cardinality care careful carries carry out carry cartesian case category theory cause caution Cayley diagram center centralizer certain certainly cf. chain ring chain chance change chapter characteristic subgroup characteristic characterization check checking chief chiefly Chinese remainder theorem choice choose circle claim class equation class number class classical classification claus clear close closed closer closure codimension coding theory coefficient coheight coincide cokernel collect column comaximal combination combinatorial combine combine come forward come from common commutative group commutative commutativity commutator commutator (sub)group commute commuting compare comparison compass complement complementary set complete completed completely completion complex complex number complexification complexity complicated component componentwise compose composite composition series composition computation compute computer science computer scientist computer conceivable concept concern concerning conclusion concrete condition conductor cone confidence confirm congruence conjecture conjugate connect connection conormal module consequence consequently conservation conserve consider consider considerable consideration consist consistency consisting of constant constitute constitutent construct constructibility constructible construction constructive contact contain containing contains content continuation continue continued fractions continuity continuous contradict contradiction contradictory contrary to convenient convention converse conversely convex convince coordinate ring coordinate system coordinate transformation coordinate corollary correction correspond to corresponding coset costly countable counted twice counted cover cover Cramer's rule criterion crossing cube current curve cut cycle cyclic cyclotomic polynomial data data structure database system database deal with decide decimal place decimal decomposable decompose decomposition decrease decrease Dedekind domain deduce deduction deed define defining definition degree dehomogenization denominator set denominator denote dense density denumerable depend on depended dependend on depth derivation derivative derive descend descending describe describe description desired detail detailed determinant determinantal ideal determine determining deterministic develop development diagram diameter dice differ difference different differentiability differentiable differential calculus differential equation differential differently (from) differing difficult difficulty digit dihedral group dimension direct factor direct limit direct product direct sum direct summand direct direction vectors direction directly disadvantage discover discrete valuation ring discrete valuation discrete discuss discussion disjunct distance distinct distribute distributive law divide divide up divisible division ring division divisor domain of definition done double line double point double dual module duality principle duality dualizing module dualizing during this during earlier easily easy edge edition effect effective element elementary embedded embedding dimension embedding employ employ empty end end endomorphism engineer enlarge enormous enter into enter entire epimorphism equality sign equality equation equivalence class equivalence relation equivalence equivalent error especially essential establish establishes estimation Euclidean Euler's totient function even number even permutation even even every evident evidently exact exactly exactness examination examine example exceedingly excellent ring excess exchange exchange exclude exercise exercise exist existence existing expand expansion experience experiment explain explicit form exploit express expression extend to extend extended extension ideal extension module extension fact factor group factor out factor factorial factorization fall familiar family feasible Fermat's little theorem Fermat's prime few fiber product fiber sum fiber field field extension field of fractions field-theoretic figure find finished finite finitely many finitely first first first of all fitting ideal fitting invariant fix fixed field fixed flat module flat follow follow following foot of perpendicular for for example for instance forego form-ring form form formal formation formula formulate formulation fraction fractional framework free frequent frequently from fulfil full fully function field function germ function fundamental theorem of algebra fundamental further furthermore future Galois extension Galois group Galois theory gather up Gaussian gcd general generalization generalize generalized generate generated generation generator generic geometric geometry get get give give up given global glue up go ahead go into Gorenstein ring graded ring graded grading great greatest common divisor ground field group structure group group table grow guarantee half halve handle happen hard have height help hence here hierarchy Hilbert polynomial Hilbert's basis theorem Hilbert's syzygy theorem Hilbert's zero theorem hint hitherto hold holomorphic homeomorphic homeomorphism homework homogeneous homological homomorphism hope for however hyperplane at infinity hyperplane hypersurface hypothesis i.e. idea ideal-theoretic ideal idempotent identify identity element identity map identity if and only if if illustrate illustration illustrative image imaginary axis imaginary immediate immediately implement implementation imply importance important improve improvement in accordance with in addition in any case in case in common in detail in fact in general in no way in order to in other words in particular in short in spite of in the following way in the following in turn in-depth include including inclusion-reversing inclusion increase increase indeed independent independent indeterminate index indexed indicate indices indirect indirectly induced induced topology induction hypothesis induction inequality inessential infer infimum infinite infinitely infinity information studies information injective injectivity inner automorphism input inquiry inseparability degree inseparable insert insert insertion inside insight instead of instruction integer integral calculus integral closure integral domain integral integrally closed integration intensify interest interesting intermediate field intermediate value theorem internal direct product interpolation intersect intersection intersection multiplicity intersection scheme introduce introduce introduction invariant invent inverse inverse inverse element inverse image inverse map invertible investigate investigation involved irreducibility irreducible is called is sufficient isomorphic isomorphism iteration Jacobian just as just kernel key role kind know know how to handle sb/sth knowledge Krull dimension Lagrange's resolvent large largest last later latest lattice point law lcm lead leading coefficient leading form least common multiple least leave out left coset left identity left inverse left multiplication left zero divisor Lemma lemma length lengthen lengthened let Let A be B Let A and B be C letter lie lies likelihood likely limit limitation line bundle line linear combination linear factor linear linearly dependent linearly independent linearly ordered literature little little local localization localize localizes locus look low magnify make a start on sth manner many map mapping mark material mathematician mathematics matrix maximal maximality maximum mean meaning meaningful meet meets memory method methodical microcomputer microprocessor middle minimal polynomial minimal minimality mistake misunderstanding model module module of fractions module of relations monic monogenic monoidal transformation monomial monotonic monotonically decreasing monotonically increasing monotonically more detailed more generally more moreover morphism mostly multiple zero multiple multiplication table multiplication multiplicative multiplicity multiply mutual mutually n-dimensional n-gon n-rowed n-tuple name namely natural number natural naturally nearly necessary necessity need need negative neighborhood neither neither A nor B net network never nilpotent nilradical Noetherian non-constant non-decomposable non-denumerable non-solvable nonnegative nonsingular nontrivial nonzero normal normal series normal subgroup normalization normalizer normally notation note note notice notice nth root nth number number of elements number of generators number of variables number theory number-theoretic number numbering numerical integration numerical observation observe obtain obvious obviously occupy occur odd permutation odd of often omit on account of which on average on principle on the basis of on the other hand one other one-one one only open open operating system operation operation table operator optimization or or order order of zero ordered ordinary orientation origin other otherwise outer automorphism outline output outside outside overview own p-group p-subgroup page pair pairwise paragraph parallel projection parallel parallelogram parametric representation part partial participating particularly partition function partition partition passage path pattern people percent percentage perfect perhaps permutation group permutation permute perpendicular phenomenon Picard group place place plane region plane plate play point at infinity point of contact point pole polynomial algebra polynomial ring polynomial positive positively possess possibility possible possibly power series ring power series power powerful practice practice precise precisely precision predominant predominantly prefix present present presentable presentation preservation preserve previous primary component primary decomposition primary ideal primary prime divisor prime element prime factorization prime field prime ideal chain prime ideal prime number primitive principal diagonal principal ideal principal or main theorem principle probability density probability distribution probability function probability probable probably problem procedure proceed process produce produce product program programming language programming progress projection projective proof proper property proportion proposition prototype prove provide purely inseparable purely transcendental purpose put in front put in order put quadratic reciprocity quadric quasicompact quasihomogeneous question quite quotient field quotient formation quotient group quotient ring quotient radical radical extension radius raise rank rather ratio rational number rational ray reach reach reader ready real real axis real reality really reason recall receive reciprocal recognize record recursion reduce to reduce reduce reduced reducible reduction redundant reference referring refinement reflexive regular regularity relation matrix relation relative topology relatively prime relevant remain remainder remaining remark remember remind repeat repeated repetition replace represent representation representative requirement residue class ring residue class resolution respectively rest restrict restriction restrictive result from result result reverse reversing revise to revised right coset right identity right inverse right multiplication right zero divisor right-angled ring ring extension ring of fractions ring structure rise rise role root of unity root rotate rough roughly row rule ruler run-time running time sample satisfy save scalar multiplication scheme section see segment selection semiconductor semicontinuity semigroup ring semigroup semilocal sense sensible sentence separability degree separable closure separable sequence series set difference set-theoretic set set setup several times several shape sharp sharpen sheaf short shorten shortening show side sign significance significant similar similar simple group simple module simple point simple simplification simplify simply simulate simultaneous since single singular point singularity situation sketch sketch small smaller smallest smooth snake lemma so-called socle software technology software solution set solution solvability solvable group solvable some somebody something sometimes sort sought for space curve space span sparse speak special speciality specialization specialize specific specification specify spectrum speed up spell spelling sphere split split off split up splits splitting field splitting-off splitting square root square stalk standstill start statement stationary stay step stick store straight line straight straightedge strategy strictly monotonic structure sheaf structure studies sub- subdeterminant subfield subgroup subject submodule subring subscheme subsequent subset subspace substitute substitute substitution subtract subtraction subvariety such that such suffice sufficient suffix suitable sum superfluous superring supplement supply support suppose supremum sure surface surjective surjectivity surplus survey Sylow group Sylow subgroup symbol symbolic symmetric system system of equations system of parameters systematic syzygy module table take into account take part take takes talk tally tangent space tangent tangential cone tangential hyperplane task technical technique tensor product term by term term terminology termwise textbook than that is that's why the same the trivial group the zero ideal the zero polynomial theorem theory there is/are therefore thin think this sort of thorough through tight till now together tool topological space topology torsion coefficient torsion group torsion module torsion subgroup torsion submodule torsion-free total totality totally totally ordered touch trace track transcendence basis transcendence degree transcendental transfer transformation transition transitive transposition treat tremendous triangle trick trisection trivial triviality try Turing machine turn turn out type unary uncountable under underlying understand uneven unimodular unimportant union unique uniqueness unit circle unit unity universal field universal unmixed unmixedness unnecessary unshortenable unusual up to use use useful usual valid validity valuation value vanish vanishing ideal vanishing variable exchange trick variable variety varying vector bundle vector verification verify vertex vertical violation volume warning way web Weierstrass' zero theorem well-defined well-known where whether whole why win with with one another with respect to within without without difficulty without loss of generality work out write yield you zero zero divisor zero element zero point zero ring zero set zero vector Zorn's lemma