Merge branch 'NumberFields' of https://github.com/Macaulay2/Workshop-…

…2024-Utah into NumberFields

NumberFields/KarlsTestingFile.m2

@@ -1,7 +1,10 @@
+restart
 loadPackage "NumberFields"
 
 R = QQ[x]/ideal(x^3-2)
 S = R[y]/ideal(y^2+y+1)
+remakeField S
+U = (flattenRing)
 T = QQ[x,y, Degrees=>{0,0}]/ideal(x^3-2,y^2+y+1)
 
 K = numberField R
@@ -15,3 +18,8 @@ pushFwd(phi)
 
 loadPackage("PushForward", DebuggingMode => true, Reload=>true)
 
+
+
+R = QQ[x,y]/ideal(y^2-x*(x-1)*(x-2), x^2-y*(y-1)*(y-2))
+psi = map(R, QQ)
+pushFwd(psi)

NumberFields/NumberFields.m2

@@ -25,8 +25,9 @@ export{
    "splittingField",
    "compositums",
    "simpleExt",
-   "remakField",--this probably shouldn't be exposed to the user long term
    "asExtensionOfBase"--this probably shouldn't be exposed to the user long term
+   "remakeField",--this probably shouldn't be exposed to the user long term
+   "minimalPolynomial"
 };
 
 NumberField = new Type of HashTable
@@ -49,12 +50,14 @@ remakeField(Ring) := opts -> R1 -> (
     local amb;
     local myIdeal;
     a := local a;
-    if instance(R1, QuotientRing) then (amb = ambient R1; myIdeal = ideal R1) else (amb = R1; myIdeal = ideal(sub(0,R1)));
+    R2 := (flattenRing R1)#0;
+    if instance(R2, QuotientRing) then (amb = ambient R2; myIdeal = ideal R2) else (amb = R2; myIdeal = ideal(sub(0,R2)));
     numVars := #(gens amb);
 
-    if opts.Variable === null then (var = local symbol a) else (var = opts.Variable);
-    newRing1 := QQ(monoid[(var_1..var_numVars)]);--we may want to think about the monomial order, we also might want to consider messing around with choosing a better presentation.
-    phi := map(R1, newRing1, gens amb);
+    if opts.Variable === null then (var = a) else (var = opts.Variable);
+    varList := {var_1..var_numVars};
+    newRing1 := QQ(monoid[varList]);--we may want to think about the monomial order, we also might want to consider messing around with choosing a better presentation.
+    phi := map(newRing1, amb, gens newRing1);
 
     newRing1/phi(myIdeal)
 )
@@ -68,37 +71,81 @@ numberField(RingElement) := opts -> f1 -> (
 
     -- Verifies that the resulting quotient is a field.
     if f1 == 0 then error("Expected nonzero polynomial.");
-    print(f1);
-    if not isPrime ideal(f1) then error("Expected an irreducible polynomial.");
+    --if not isPrime ideal(f1) then error("Expected an irreducible polynomial.");
 
-    new NumberField from { 
-        ring => toField (R1/ideal(f1)), 
-        cache => new CacheTable from {}
-    }
+    numberField(R1/ideal(f1))
 )
 
+
+
 numberField(Ring) := opts -> R1 -> (
-    if R1===QQ then return new NumberField from {ring => R1};
+    if R1===QQ then return new NumberField from {
+            ring => R1, 
+            pushFwd => pushFwd(map(QQ,QQ)),
+            cache => new CacheTable from {}
+        };
 
     if not isPrime (ideal 0_R1) then error("Expected a field.");
     if not dim R1 == 0 then error("Expected a field.");
 
     if char R1 != 0 then error("Expected characteristic 0.");
-
-    flattenedR1 := (flattenRing(R1))#0;
-    iota := map(flattenedR1,QQ);
-    try pushFwd(iota) else error("Not finite dimensional over QQ");
+    outputRing := remakeField(R1);
+    iota := map(outputRing,QQ);
+    local myPushFwd;
+    try myPushFwd = pushFwd(iota) else error("Not finite dimensional over QQ");
+    genMinPolys := apply(gens outputRing, h->minimalPolynomial(h));
+
+    genList := gens outputRing;
+    genCt := #genList;
+    i := 0;
+    deg := degree outputRing;
+
+    myStr := "Number Field, degree ";
+    myStr = myStr | toString(deg) | " / Q, generated by:";
+    if genCt == 0 then myStr = myStr + " nothing";
+    while (i < genCt) do (
+        myStr = myStr | " " | toString(genList#i) | " (" | toString (genMinPolys#i) | ")";
+        i = i+1;
+        if (i < genCt) then myStr = myStr | ", ";
+    );
 
     new NumberField from {
-        ring => toField (flattenedR1), 
-        cache => new CacheTable from {}
+        ring => toField outputRing, 
+        pushFwd => myPushFwd,
+        String => myStr,
+        minimalPolynomial => genMinPolys, --a list of min polys of gens, for display purposes
+        cache => new CacheTable from {degree => deg}
     }
 )
 
 internalNumberFieldConstructor := R1 -> (
 
 );
 
+--*****************************
+--NumberField display
+--*****************************
+
+
+net NumberField := nf -> (
+    -*
+    myStr := "Number Field, degree ";
+    myStr = myStr | toString(degree nf) | " / Q, generated by:";
+    i := 0;
+    genCt := #(gens ring nf);
+    genList := gens ring nf;
+    if genCt == 0 then myStr = myStr + " nothing";
+    while (i < genCt) do (
+        myStr = myStr | " " | toString(genList#i) | " (" | toString (nf#minimalPolynomial#i) | ")";
+        i = i+1;
+        if (i < genCt) then myStr = myStr + ", ";
+    );
+    --do some code
+    *-
+
+    nf#String
+)
+
 --****************************
 --NumberField basic operations
 --****************************
@@ -156,6 +203,14 @@ degree(NumberFieldExtension) := nfe -> (
     rk
 )
 
+norm(RingElement) := (elt) ->(
+    S := ring elt;
+    return det pushFwd(map(S^1, S^1, {{elt}}));
+);
+trace(RingElement) := (elt) -> (
+    S := ring elt;
+    return trace pushFwd(map(S^1, S^1, {{elt}}));
+);
 --*************************
 --Methods
 --*************************
@@ -179,6 +234,7 @@ isGalois(NumberFieldExtension) := opts -> iota -> (
 isGalois(RingMap) := opts -> iota -> (
    isGalois(numberFieldExtension iota)
 )
+
 -- splittingField method
 splittingField = method(Options => {})
 splittingField(RingElement) := opts -> f1 -> (
@@ -221,21 +277,44 @@ simpleExt(NumberField) := opts -> nf ->(
     D := degree K;
     --We find an element that produces a degree D field extension.
     d := 0;
+    c := 0;
     while d < D do 
     (
         r := random(1, K);
         xx := local xx;
         R := QQ[xx];
         phi := map( K, R, {r});
         if  isPrime (kernel phi) then (
-            I := kernel phi *sub (( 1/(((coefficients (first entries gens kernel phi)_0)_1)_0)_0), QQ);
+            I := kernel phi *sub (( 1/(((coefficients (first entries gens kernel phi)_0)_1)_0)_0), R);
             simpleExt := numberField(R / I);
-            d := degree simpleExt;
+            d = degree simpleExt;
         );
     );
     return simpleExt;
 )
 
+minimalPolynomial = method(Options => {})
+minimalPolynomial(RingElement) := opts -> f1 -> (
+    R1 := ring f1;
+    S1 := (coefficientRing(R1))[local y];
+    P1 := pushFwd(map(R1, coefficientRing(R1)));
+    A1 := (P1#2)(1_R1);
+    pow1 := 1;
+    while gens(kernel(A1))==0 do (
+        A1 = A1|(P1#2)(f1^pow1);
+        pow1 += 1;
+    );
+    M1 := matrix({{1_S1}});
+    for i1 from 1 to (pow1-1) do (
+        M1 |= y^i1;
+    );
+    (entries (M1*(gens(kernel(A1)))))#0#0
+)
+minimalPolynomial(List) := opts -> L1 -> (
+    apply(L1, i -> minimalPolynomial(i))
+)
+
+
 --*****************************
 --Documentation
 --*****************************

NumberFields/minimalPolynomial.m2

@@ -0,0 +1,64 @@
+loadPackage ("NumberFields", Reload=>true)
+
+-- Nicholas' minimal polynomial file.
+-- Method to compute minimal polynomial via pushFwd.
+-- Eventually extend to also compute minimal polynomials for a list of elements.
+minimalPolynomialViaPushFwd = method(Options => {})
+minimalPolynomialViaPushFwd(RingElement) := opts -> f1 -> (
+    R1 := ring f1;
+    S1 := (coefficientRing(R1))[y];
+    P1 := pushFwd(map(R1, QQ));
+    A1 := (P1#2)(1_R1);
+    pow1 := 1;
+    while not gens(kernel(A1))==0 do (
+        A1 |= (P1#2)(f1^pow1);
+        pow1 += 1;
+    );
+    M := matrix({{1_S1}});
+    for i1 from 1 to pow1 do (
+        M |= y^i1;
+    );
+    (entries (M*K))#0#0
+)
+
+loadPackage ("NumberFields", Reload=>true)
+R1 = QQ[x]/(x^3-2)
+S1 = (coefficientRing(R1))[y]
+elt1 = x
+--phi1 = map(R1, QQ)
+P1 = pushFwd(map(R1, QQ))
+g1 = (P1#2)(1_R1)
+g2 = (P1#2)(elt1)
+g3 = (P1#2)(elt1^2)
+g4 = (P1#2)(elt1^3)
+A = g1|g2|g3|g4
+K = gens(kernel(A))
+M = matrix({{1, y, y^2, y^3}})
+(entries (M*K))#0#0
+
+--****************
+--Tests
+--****************
+
+loadPackage ("NumberFields", Reload=>true)
+-- This one should return y^3 - 2
+R1 = QQ[x]/(x^3-2)
+f = x
+minimalPolynomial f
+
+-- This one should return y^3 - 4
+f = x^2
+minimalPolynomial f
+
+-- This one should return y^3 - 6*y - 6
+f = x^2+x
+minimalPolynomial f
+
+minimalPolynomial {x, x^2, x^2+x}
+
+-- This one should return y^4 - 2*y^2 + 9
+R1 = QQ[x,y]/(x^2+1,y^2-2)
+f = x+y
+minimalPolynomial f
+
+simpleExt numberField R1

NumberFields/norm.m2

@@ -3,4 +3,6 @@ norm(RingElement) := (elt) ->(
     S = ring elt;
     return det pushFwd(map(S^1, S^1, {{elt}}));
 );
+R = QQ[x]/(x^3-2)
+norm(1_R)
 

NumberFields/splittingField.m2

@@ -108,4 +108,95 @@ S = splittingField f1
 T = (ring target S)[y]
 f11 = (map(T, R, {y})) f1
 L2 = decompose ideal f11
-polynomialSplits f11
+polynomialSplits f11
+
+-- Let's try to see what's going on here.
+R1 = ring f1
+S1 = R1;
+K1 = coefficientRing R1
+K2 = coefficientRing R1
+variableIndex = 1
+-- Begin for loop
+L1 = decompose ideal f1
+currentEntry = (entries gens L1#0)#0
+length(currentEntry)==1 and max(degree(currentEntry#0))==1
+degree(currentEntry#0)
+K1 = R1/(L1#0)
+describe K1
+S1 = K1[a_variableIndex]
+describe S1
+phi1 = map(S1, R1, {a_variableIndex})
+f1 = phi1(f1)
+f1 = f1//(a_variableIndex-x)
+R1 = S1
+describe R1
+describe K1
+variableIndex += 1
+-- Second iteration of for loop
+L1 = decompose ideal f1
+currentEntry = (entries gens L1#0)#0
+K1 = R1/(L1#0)
+describe K1
+
+-- My simpleExt idea
+(flattenRing K1)#0
+NF1 = numberField K1
+NF2 = simpleExt NF1
+K1 = ring NF2
+
+-- Back to the original loop
+S1 = K1[a_variableIndex]
+describe S1
+phi1 = map(S1, R1, {a_variableIndex})
+f1 = phi1(f1)
+f1 //= (a_variableIndex-a_(variableIndex-1))
+R1 = S1
+describe R1
+describe K1
+variableIndex += 1
+
+--(flattenRing K1)#0
+--NF1 = numberField K1
+--NF2 = simpleExt NF1
+--K2 = ring NF2
+--L1 = decompose ideal f1
+--L1#0
+--describe S1
+
+-- Third iteration of for loop
+L1 = decompose ideal f1
+currentEntry = (entries gens L1#0)#0
+degree currentEntry#0
+K1 = R1/(L1#0)
+describe K1
+
+-- Another simpleExt
+NF1 = numberField K1
+NF2 = simpleExt NF1
+K1 = ring NF2
+
+S1 = K1[a_variableIndex]
+describe S1
+phi1 = map(S1, R1, {a_variableIndex})
+f1 = phi1(f1)
+f1 //= (a_variableIndex-a_(variableIndex-1))
+R1 = S1
+describe R1
+describe K1
+variableIndex += 1
+-- Fourth iteration of for loop
+-- Note: the below line is where things take too long. Need to figure out a way to get decompose running more quickly.
+L1 = decompose ideal f1
+currentEntry = (entries gens L1#0)#0
+degree currentEntry#0
+K1 = R1/(L1#0)
+describe K1
+S1 = K1[a_variableIndex]
+describe S1
+phi1 = map(S1, R1, {a_variableIndex})
+f1 = phi1(f1)
+f1 //= (a_variableIndex-a_(variableIndex-1))
+R1 = S1
+describe R1
+describe K1
+variableIndex += 1

NumberFields/trace.m2

@@ -3,3 +3,6 @@ trace(RingElement) := (elt) -> (
     S = ring elt;
     return trace pushFwd(map(S^1, S^1, {{elt}}));
 );
+R = QQ[x]/(x^3-2)
+trace(x)
+