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# integralClosure(...,Verbosity=>...) -- display a certain amount of detail about the computation

- Usage:
    `integralClosure(R, Verbosity => n)`
- Inputs:
  - n, an integer, The higher the number, the more information is displayed. A value of 0 means: keep quiet.

## Description

When the computation takes a considerable time, this function can be used to decide if it will ever finish, or to get a feel for what is happening during the computation.

    i1 : R = QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3);

    i2 : time R' = integralClosure(R, Verbosity => 2)
     [jacobian time .0005489 sec #minors 3]
    integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2

     [step 0:
          radical (use minprimes) .00282466 seconds
          idlizer1:  .00782705 seconds
          idlizer2:  .00866467 seconds
          minpres:   .00838684 seconds
      time .0394431 sec  #fractions 4]
     [step 1:
          radical (use minprimes) .00234045 seconds
          idlizer1:  .0118696 seconds
          idlizer2:  .0103475 seconds
          minpres:   .0117409 seconds
      time .0473824 sec  #fractions 4]
     [step 2:
          radical (use minprimes) .00230946 seconds
          idlizer1:  .0122185 seconds
          idlizer2:  .00984464 seconds
          minpres:   .00940833 seconds
      time .0448876 sec  #fractions 5]
     [step 3:
          radical (use minprimes) .00229545 seconds
          idlizer1:  .0734378 seconds
          idlizer2:  .0125437 seconds
          minpres:   .0150642 seconds
      time .115135 sec  #fractions 5]
     [step 4:
          radical (use minprimes) .0022606 seconds
          idlizer1:  .00848334 seconds
          idlizer2:  .0159713 seconds
          minpres:   .0114038 seconds
      time .0501376 sec  #fractions 5]
     [step 5:
          radical (use minprimes) .00229417 seconds
          idlizer1:  .00814519 seconds
      time .0170743 sec  #fractions 5]
     -- used 0.317865s (cpu); 0.285355s (thread); 0s (gc)

    o2 = R'

    o2 : QuotientRing

    i3 : trim ideal R'

                         3   2                     2 2    4           4
    o3 = ideal (w   z - x , w   x - w   , w   x - y z  - z  - z, w   x  - w   z,
                 4,0         4,0     1,1   1,1                    4,0      1,1
         ------------------------------------------------------------------------
                     2 2     2 3    2   3      2   3 2      4 2      2 4       2
         w   w    - x y z - x z  - x , w    + w   x y  - x*y z  - x*y z  - 2x*y z
          4,0 1,1                       4,0    4,0
         ------------------------------------------------------------------------
              3           3    2      6 2    6 2
         - x*z  - x, w   x  - w    + x y  + x z )
                      4,0      1,1

    o3 : Ideal of QQ[w   , w   , x..z]
                      4,0   1,1

    i4 : icFractions R

           3   2 2    4
          x   y z  + z  + z
    o4 = {--, -------------, x, y, z}
           z        x

    o4 : List

## Caveat

The exact information displayed may change.

## Functions with optional argument named Verbosity:

- icFracP(...,Verbosity=>...) -- Prints out the conductor element and the number of intermediate modules it computed.
- `idealizer(...,Verbosity=>...)` -- see idealizer -- compute Hom(I,I) as a quotient ring
- integralClosure(...,Verbosity=>...) -- display a certain amount of detail about the computation
- `isPrime(Ideal,Verbosity=>...)` -- see isPrime(Ideal) -- whether an ideal is prime
- `makeS2(...,Verbosity=>...)` -- see makeS2 -- compute the S2ification of a reduced ring
- `decompose(Ideal,Verbosity=>...)` -- see minimalPrimes -- minimal primes of an ideal
- `minimalPrimes(...,Verbosity=>...)` -- see minimalPrimes -- minimal primes of an ideal
- `ringFromFractions(...,Verbosity=>...)` -- see ringFromFractions -- find presentation for f.g. ring

## Further information

- Default value: 0
- Function: integralClosure -- integral closure of an ideal or a domain
- Option key: Verbosity (missing documentation)

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/IntegralClosure.m2:1453:0.