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Welcome to the OSCAR project, a visionary new computer algebra system which combines the capabilities of four cornerstone systems: GAP, Polymake, Antic and Singular.

OSCAR requires Julia 1.6 or newer. In principle it can be installed and used like any other Julia package; doing so will take a couple of minutes:

```
julia> using Pkg
julia> Pkg.add("Oscar")
julia> using Oscar
```

However, some of OSCAR's components have additional requirements. For more detailed information, please consult the installation instructions on our website.

Please read the introduction for new developers in the OSCAR manual to learn more on how to contribute to OSCAR.

```
julia> using Oscar
___ ____ ____ _ ____
/ _ \ / ___| / ___| / \ | _ \ | Combining ANTIC, GAP, Polymake, Singular
| | | |\___ \| | / _ \ | |_) | | Type "?Oscar" for more information
| |_| | ___) | |___ / ___ \| _ < | Manual: https://docs.oscar-system.org
\___/ |____/ \____/_/ \_\_| \_\ | Version 1.2.0-DEV
julia> k, a = quadratic_field(-5)
(Imaginary quadratic field defined by x^2 + 5, sqrt(-5))
julia> zk = maximal_order(k)
Maximal order of Imaginary quadratic field defined by x^2 + 5
with basis AbsSimpleNumFieldElem[1, sqrt(-5)]
julia> factorizations(zk(6))
2-element Vector{Fac{AbsSimpleNumFieldOrderElem}}:
-1 * -3 * 2
-1 * (-sqrt(-5) - 1) * (-sqrt(-5) + 1)
julia> Qx, x = polynomial_ring(QQ, [:x1,:x2])
(Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x1, x2])
julia> R = grade(Qx, [1,2])[1]
Multivariate polynomial ring in 2 variables over QQ graded by
x1 -> [1]
x2 -> [2]
julia> f = R(x[1]^2+x[2])
x1^2 + x2
julia> degree(f)
[2]
julia> F = free_module(R, 1)
Free module of rank 1 over R
julia> s = sub(F, [f*F[1]])[1]
Submodule with 1 generator
1 -> (x1^2 + x2)*e[1]
represented as subquotient with no relations.
julia> H, mH = hom(s, quo(F, s)[1])
(hom of (s, Subquotient of
1 -> e[1]
by
1 -> (x1^2 + x2)*e[1]), Map: H -> set of all homomorphisms from s to subquotient of Submodule with 1 generator
1 -> e[1]
by Submodule with 1 generator
1 -> (x1^2 + x2)*e[1])
julia> mH(H[1])
Map with following data
Domain:
=======
Submodule with 1 generator
1 -> (x1^2 + x2)*e[1]
represented as subquotient with no relations.
Codomain:
=========
Subquotient of Submodule with 1 generator
1 -> e[1]
by Submodule with 1 generator
1 -> (x1^2 + x2)*e[1]
```

Of course, the cornerstones are also available directly:

```
julia> C = Polymake.polytope.cube(3);
julia> C.F_VECTOR
pm::Vector<pm::Integer>
8 12 6
julia> RP2 = Polymake.topaz.real_projective_plane();
julia> RP2.HOMOLOGY
pm::Array<topaz::HomologyGroup<pm::Integer> >
({} 0)
({(2 1)} 0)
({} 0)
```

If you have used OSCAR in the preparation of a paper please cite it as described below:

```
[OSCAR]
OSCAR -- Open Source Computer Algebra Research system, Version 1.2.0-DEV,
The OSCAR Team, 2024. (https://www.oscar-system.org)
[OSCAR-book]
Wolfram Decker, Christian Eder, Claus Fieker, Max Horn, Michael Joswig, eds.
The Computer Algebra System OSCAR: Algorithms and Examples,
Algorithms and Computation in Mathematics, Springer, 2024.
```

If you are using BibTeX, you can use the following BibTeX entries:

```
@misc{OSCAR,
key = {OSCAR},
organization = {The OSCAR Team},
title = {OSCAR -- Open Source Computer Algebra Research system,
Version 1.2.0-DEV},
year = {2024},
url = {https://www.oscar-system.org},
}
@book{OSCAR-book,
editor = {Decker, Wolfram and Eder, Christian and Fieker, Claus and Horn, Max and Joswig, Michael},
title = {The {C}omputer {A}lgebra {S}ystem {OSCAR}: {A}lgorithms and {E}xamples},
year = {2024},
publisher = {Springer},
series = {Algorithms and {C}omputation in {M}athematics},
volume = {32},
edition = {1},
url = {https://link.springer.com/book/9783031621260},
month = {8},
issn = {1431-1550},
}
```

The development of this Julia package is supported by the Deutsche Forschungsgemeinschaft DFG within the Collaborative Research Center TRR 195.