[was@modular 03-27-02]$ [was@modular 03-27-02]$ Magma V2.8-10     Wed Mar 27 2002 17:56:01 on modular  [Seed = 3394549903]
Type ? for help.  Type <Ctrl>-D to quit.

Loading startup file "/home/was/magma/local/emacs.m"

Loading "/home/was/magma/local/init.m"
> K := NumberField(x^2+1);
> R<y> := PolynomialRing(K);
> R;
Univariate Polynomial Ring in y over K
> L := NumberField(y^2+7);
> L;
Number Field with defining polynomial y^2 + 7 over K
> Degree(L);
2
> AbsoluteDegree(L);
4
> S<z> := PolynomialRing(L);
> 
> W := NumberField(z^2+1);

>> W := NumberField(z^2+1);
                   ^
Runtime error in 'NumberField': Argument 1 is not irreducible
> Factorization(z^2+1);
[
<z - K.1, 1>,
<z + K.1, 1>Cyclo
]
> Factorization(z^2+y);
[
<z, 1>,
<z + 1, 1>
]
> 
> Qp := pAdicField(37);R<x> := PolynomialRing(Qp);f :=  R!CyclotomicPolynomial(37);g := Evaluate(f,x+1);
> 
> f;
x^36 + x^35 + x^34 + x^33 + x^32 + x^31 + x^30 + x^29 + x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
> 
 g;
> x^36 + 37*x^35 + 18*37*x^34 + 210*37*x^33 + 1785*37*x^32 + 11781*37*x^31 + 62832*37*x^30 + 278256*37*x^29 + 1043460*37*x^28 + 3362260*37*x^27 + 9414328*37*x^26 + 23107896*37*x^25 + 50067108*37*x^24 + 96282900*37*x^23 + 165056400*37*x^22 + 253086480*37*x^21 + 347993910*37*x^20 +    q429874830*37*x^19 + 477638700*37*x^18 + 477638700*37*x^17 + 429874830*37*x^16 + 347993910*37*x^15 + 253086480*37*x^14 + 165056400*37*x^13 + 96282900*37*x^12 + 50067108*37*x^11 + 23107896*37*x^10 + 9414328*37*x^9 + 3362260*37*x^8 + 1043460*37*x^7 + 278256*37*x^6 + 62832*37*x^5 + 11781*37*x^4 + 1785*37*x^3 + 210*37*x^2 + 18*37*x + 37
> x^36 + 37*x^35 + 18*37*x^34 + 210*37*x^33 + 1785*37*x^32 + 11781*37*x^31 + 62832*37*x^30 + 278256*37*x^29 + 1043460*37*x^28 + 3362260*37*x^27 + 9414328*37*x^26 + 23107896*37*x^25 + 50067108*37*x^24 + 96282900*37*x^23 + 165056400*37*x^22 + 253086480*37*x^21 + 347993910*37*x^20 +    q429874830*37*x^19 + 477638700*37*x^18 + 477638700*37*x^17 + 429874830*37*x^16 + 347993910*37*x^15 + 253086480*37*x^14 + 165056400*37*x^13 + 96282900*37*x^12 + 50067108*37*x^11 + 23107896*37*x^10 + 9414328*37*x^9 + 3362260*37*x^8 + 1043460*37*x^7 + 278256*37*x^6 + 62832*37*x^5 + 11781*37*x^4 + 1785*37*x^3 + 210*37*x^2 + 18*37*x + 37


> > > 
> 
> 
> g;

>> g;
   ^
User error: bad syntax
> g := Evaluate(f,x+1);
> g;
x^36 + 37*x^35 + 18*37*x^34 + 210*37*x^33 + 1785*37*x^32 + 11781*37*x^31 + 62832*37*x^30 + 278256*37*x^29 + 1043460*37*x^28 + 3362260*37*x^27 + 9414328*37*x^26 + 23107896*37*x^25 + 50067108*37*x^24 + 96282900*37*x^23 + 165056400*37*x^22 + 253086480*37*x^21 + 347993910*37*x^20 + 429874830*37*x^19 + 477638700*37*x^18 + 477638700*37*x^17 + 429874830*37*x^16 + 347993910*37*x^15 + 253086480*37*x^14 + 165056400*37*x^13 + 96282900*37*x^12 + 50067108*37*x^11 + 23107896*37*x^10 + 9414328*37*x^9 + 3362260*37*x^8 + 1043460*37*x^7 + 278256*37*x^6 + 62832*37*x^5 + 11781*37*x^4 + 1785*37*x^3 + 210*37*x^2 + 18*37*x + 37
> K := TotallyRamifiedExtension(Qp, g) ;
> K;
Local Field with Eisenstein polynomial x^36 + 37*x^35 + 666*x^34 + 7770*x^33 + 66045*x^32 + 435897*x^31 + 2324784*x^30 + 10295472*x^29 + 38608020*x^28 + 124403620*x^27 + 348330136*x^26 + 854992152*x^25 + 1852482996*x^24 + 3562467300*x^23 + 6107086800*x^22 + 9364199760*x^21 + 12875774670*x^20 + 15905368710*x^19 + 17672631900*x^18 + 17672631900*x^17 + 15905368710*x^16 + 12875774670*x^15 + 9364199760*x^14 + 6107086800*x^13 + 3562467300*x^12 + 1852482996*x^11 + 854992152*x^10 + 348330136*x^9 + 124403620*x^8 + 38608020*x^7 + 10295472*x^6 + 2324784*x^5 + 435897*x^4 + 66045*x^3 + 7770*x^2 + 666*x + 37 over Qp
> S<y> := PolynomialRing(K);
> /home/was/bin/me: line 4: 30862 Terminated              magma2.8
[was@modular 03-27-02]$ me
Magma V2.8-10     Wed Mar 27 2002 18:20:31 on modular  [Seed = 1944919287]
Type ? for help.  Type <Ctrl>-D to quit.
Qp := pAdicField(37);R<x> := PolynomialRing(Qp);f :=  R!CyclotomicPolynomial(37);g := Evaluate(f,x+1);

Loading startup file "/home/was/magma/local/emacs.m"

Loading "/home/was/magma/local/init.m"
> > 
> 
> g;
x^36 + 37*x^35 + 18*37*x^34 + 210*37*x^33 + 1785*37*x^32 + 11781*37*x^31 + 62832*37*x^30 + 278256*37*x^29 + 1043460*37*x^28 + 3362260*37*x^27 + 9414328*37*x^26 + 23107896*37*x^25 + 50067108*37*x^24 + 96282900*37*x^23 + 165056400*37*x^22 + 253086480*37*x^21 + 347993910*37*x^20 + 429874830*37*x^19 + 477638700*37*x^18 + 477638700*37*x^17 + 429874830*37*x^16 + 347993910*37*x^15 + 253086480*37*x^14 + 165056400*37*x^13 + 96282900*37*x^12 + 50067108*37*x^11 + 23107896*37*x^10 + 9414328*37*x^9 + 3362260*37*x^8 + 1043460*37*x^7 + 278256*37*x^6 + 62832*37*x^5 + 11781*37*x^4 + 1785*37*x^3 + 210*37*x^2 + 18*37*x + 37
> 
> K;
Local Field with Eisenstein polynomial x^36 + 37*x^35 + 666*x^34 + 7770*x^33 + 66045*x^32 + 435897*x^31 + 2324784*x^30 + 10295472*x^29 + 38608020*x^28 + 124403620*x^27 + 348330136*x^26 + 854992152*x^25 + 1852482996*x^24 + 3562467300*x^23 + 6107086800*x^22 + 9364199760*x^21 + 12875774670*x^20 + 15905368710*x^19 + 17672631900*x^18 + 17672631900*x^17 + 15905368710*x^16 + 12875774670*x^15 + 9364199760*x^14 + 6107086800*x^13 + 3562467300*x^12 + 1852482996*x^11 + 854992152*x^10 + 348330136*x^9 + 124403620*x^8 + 38608020*x^7 + 10295472*x^6 + 2324784*x^5 + 435897*x^4 + 66045*x^3 + 7770*x^2 + 666*x + 37 over Qp
> K := TotallyRamifiedExtension(Qp, g) ;S<y> := PolynomialRing(K);
> Factorization(y^2+1);
[
<(1 + O(K.1^20))*y - 6 + O(K.1^20), 1>,
<(1 + O(K.1^20))*y + 6 + O(K.1^20), 1>
]
> Factorization(y^5+17);
[
<(1 + O(K.1^20))*y + 5 + O(K.1^20), 1>,
<(1 + O(K.1^20))*y^4 - (5 + O(K.1^20))*y^3 - (12 + O(K.1^20))*y^2 - (14 + O(K.1^20))*y - 4 + O(K.1^20), 1>
]
> Factorization(y^19+7*y^8+3);



[Interrupt twice in half a second; exiting]

Total time: 21.020 seconds
[was@modular 03-27-02]$ [was@modular 03-27-02]$ [was@modular 03-27-02]$ 
[was@modular 03-27-02]$ 
[was@modular 03-27-02]$ 
[was@modular 03-27-02]$ me
Magma V2.8-10     Wed Mar 27 2002 18:21:59 on modular  [Seed = 2078610217]
Type ? for help.  Type <Ctrl>-D to quit.
:= PolynomialRing(Qp);f :=  R!CyclotomicPolynomial(37);g := Evaluate(f,x+1);

Loading startup file "/home/was/magma/local/emacs.m"

Loading "/home/was/magma/local/init.m"
> 
>> := PolynomialRing(Qp);f :=  R!CyclotomicPolynomial(37);g := Evaluate(f,x+1)
   ^
User error: bad syntax
> Qp := pAdicField(37);R<x> := PolynomialRing(Qp);f :=  R!CyclotomicPolynomial(37);g := Evaluate(f,x+1);
> S<y> := PolynomialRing(K);

>> S<y> := PolynomialRing(K);
                          ^
User error: Identifier 'K' has not been declared or assigned
> K := TotallyRamifiedExtension(Qp, g) ;S<y> := PolynomialRing(K);
> S;
Univariate Polynomial Ring in y over K
> Factorization(y^5+17*K.1*y+3);
[
<(1 + O(K.1^20))*y - 9 - 3*K.1 + K.1^2 + 12*K.1^3 - 13*K.1^5 + 2*K.1^6 + 12*K.1^7 - 17*K.1^8 - 17*K.1^10 + 13*K.1^11 + 6*K.1^13 - 16*K.1^15 + 13*K.1^16 - 12*K.1^18 + O(K.1^20), 1>,
<(1 + O(K.1^20))*y^4 + (9 + 3*K.1 - K.1^2 - 12*K.1^3 + 13*K.1^5 - 2*K.1^6 - 12*K.1^7 + 17*K.1^8 + 17*K.1^10 - 13*K.1^11 - 6*K.1^13 + 16*K.1^15 - 13*K.1^16 + 12*K.1^18 + O(K.1^20))*y^3 + (7 + 17*K.1 - 9*K.1^2 + 3*K.1^4 - K.1^5 + K.1^6 + 5*K.1^7 - 11*K.1^9 - 11*K.1^10 - 13*K.1^12 - 18*K.1^14 + K.1^15 + 15*K.1^17 + 3*K.1^19 + O(K.1^20))*y^2 + (-11 - 11*K.1 - 17*K.1^3 + 17*K.1^4 + 14*K.1^5 - 12*K.1^6 - 4*K.1^8 - 13*K.1^9 + 14*K.1^10 + 7*K.1^11 + 18*K.1^13 - 16*K.1^14 + 16*K.1^16 + 14*K.1^18 - 7*K.1^19 + O(K.1^20))*y + 12 - 4*K.1 + 15*K.1^2 - 10*K.1^3 + 12*K.1^4 + 14*K.1^5 + 11*K.1^7 + 13*K.1^8 + 9*K.1^9 - 3*K.1^10 + 7*K.1^12 - 17*K.1^13 - 6*K.1^15 - 3*K.1^17 + 16*K.1^18 + O(K.1^20), 1>
]
>