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-- Tutorial 1: Rings, Fields, and Ideals                            --
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-- Use C-x 2 to get a horizontally split window or 
-- Use C-x 3 to get a vertically split window.

-- Select the lower (or right) window and Hit F12 to start M2.

-- Hit F11 to send code from this screen to M2.

-- for help with M2 type (works in version 1.1)

viewHelp

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-- Basic Rings and Fields                                           --
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-- How do you create the integers?



-- How do you create the rational numbers?



-- How do you create a finite field of order 2?



-- Give another way to create the same finite field of order 2.



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-- Polynomial Rings                                                 --
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-- How do you create a polynomial ring in a single variable with
-- coefficients in the finite field of order 3?



-- How do you create a polynomial ring with coefficients in your
-- favorite field, in variables x, y, and z?



-- How do you create a polynomial ring with coefficients in your
-- favorite field, in variables x_1,...,x_100?  

--Hint: run
restart



-- Factor the polynomial x^8 + 128x^7 - 234x^4 + x^2 + x + 1 in a ring
-- of characteristic 7, 11, 13, and 17.



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-- Quotient Rings                                                   --
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-- Create a polynomial ring over a finite field with 101 elements with
-- three variables x, y, and z.  Call this ring A.  Create the ideal 
-- I = (xy, z^2).



-- Now compute the quotient ring B = A/I.  



-- What is the image of the polynomial x^7 + x^3*y^6 - y^10+z^4 + 1 in
-- B?



-- Let:

A = QQ[x,y,z]
B = QQ[u,v]

-- Create a ring homomorphism phi : A -> B 
-- where x -> u^2, y -> uv, z -> v^2.



-- Compute the kernel of phi.  Can M2 compute the image of this map?



-- Construct the Gaussian integers.



-- Construct a ring k[x,x^{-1}], where k is a field of your choice. 



-- Construct a ring containing nontrivial idempotents. 



-- Create a ring of characteristic p, where p is your favorite prime,
-- and use it to define the Frobenius endomorphism.



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-- Ideals                                                           --
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-- Now let's work in the ring ZZ[x,y]. 

-- Is (x+y, x-y) = (x, y)?  Compare to your answer in Assignment 1.



-- Is (x+xy, y + xy, x^2, y^2) = (x, y)?  Compare to your answer in
-- Assignment 1.



-- Is (2x^2+3y^2-11, x^2-y^2-3) = (x^2-4,y^2-1)?  Compare to your
-- answer in Assignment 1.



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-- A more complex ring                                              --
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-- Now let k be your favorite field, and let A be the ring of
-- polynomials in x over k such that the polynomials contain no terms
-- in x or x^2.  We will lead you through how to construct this ring.

-- Construct a polynomial ring where all the terms of the polynomials
-- have even degrees.  
-- See: Macaulay2Doc > rings > graded and multigraded polynomial rings



-- Construct a polynomial ring containing no terms in x.  Hint: Think
-- about quotient rings and the problem above.



-- Finally, construct a polynomial ring containing no terms in x or
-- x^2.  When constructing your ideal containing the relations, think
-- about how you obtain a 5th degree polynomial, a 6th degree
-- polynomial, and so on.



-- Now computationally check the following question: 

-- If I = (x^3, x^4), is x^5 \in I? Is (x^5)I \subset I?



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-- FIN                                                              --
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