sagemathgh-40019: Hash fraction_field_elements more appropriately
    
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to calculate 1 + 2". -->
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Equip various domains with a `canoncial_associate`, which finds unique
representatives in some rings modulo the action of the unit group. For
fields this is easy: there are only two orbits: 0 and the units (where 1
is a representative) and in ZZ we can just use "abs". For polynomial
rings we can extend representatives by normalizing the leading
coefficient to a canonical associate.

The normalization is important to find unique representatives of
fraction field elements (because multplying both numerator and
denominator by a unit doesn't change the represented class), which is
necessary for hashing them. The library uses hashes of fraction field
elements quite a bit, but luckily only in cases where we can actually
normalize denominators (in fact, "reduce" always seems to fail for such
rings, so the hash would already error out)

Fixes sagemath#35238

### 📝 Checklist

<!-- Put an `x` in all the boxes that apply. -->

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [x] I have updated the documentation and checked the documentation
preview.
    
URL: sagemath#40019
Reported by: nbruin
Reviewer(s): Martin Rubey

sagemathgh-40019: Hash fraction_field_elements more appropriately
    
<!-- ^ Please provide a concise and informative title. -->
<!-- ^ Don't put issue numbers in the title, do this in the PR
description below. -->
<!-- ^ For example, instead of "Fixes sagemath#12345" use "Introduce new method
to calculate 1 + 2". -->
<!-- v Describe your changes below in detail. -->
<!-- v Why is this change required? What problem does it solve? -->
<!-- v If this PR resolves an open issue, please link to it here. For
example, "Fixes sagemath#12345". -->

Equip various domains with a `canoncial_associate`, which finds unique
representatives in some rings modulo the action of the unit group. For
fields this is easy: there are only two orbits: 0 and the units (where 1
is a representative) and in ZZ we can just use "abs". For polynomial
rings we can extend representatives by normalizing the leading
coefficient to a canonical associate.

The normalization is important to find unique representatives of
fraction field elements (because multplying both numerator and
denominator by a unit doesn't change the represented class), which is
necessary for hashing them. The library uses hashes of fraction field
elements quite a bit, but luckily only in cases where we can actually
normalize denominators (in fact, "reduce" always seems to fail for such
rings, so the hash would already error out)

Fixes sagemath#35238

### 📝 Checklist

<!-- Put an `x` in all the boxes that apply. -->

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [x] I have updated the documentation and checked the documentation
preview.
    
URL: sagemath#40019
Reported by: nbruin
Reviewer(s): Martin Rubey

sagemathgh-40019: Hash fraction_field_elements more appropriately
    
<!-- ^ Please provide a concise and informative title. -->
<!-- ^ Don't put issue numbers in the title, do this in the PR
description below. -->
<!-- ^ For example, instead of "Fixes sagemath#12345" use "Introduce new method
to calculate 1 + 2". -->
<!-- v Describe your changes below in detail. -->
<!-- v Why is this change required? What problem does it solve? -->
<!-- v If this PR resolves an open issue, please link to it here. For
example, "Fixes sagemath#12345". -->

Equip various domains with a `canoncial_associate`, which finds unique
representatives in some rings modulo the action of the unit group. For
fields this is easy: there are only two orbits: 0 and the units (where 1
is a representative) and in ZZ we can just use "abs". For polynomial
rings we can extend representatives by normalizing the leading
coefficient to a canonical associate.

The normalization is important to find unique representatives of
fraction field elements (because multplying both numerator and
denominator by a unit doesn't change the represented class), which is
necessary for hashing them. The library uses hashes of fraction field
elements quite a bit, but luckily only in cases where we can actually
normalize denominators (in fact, "reduce" always seems to fail for such
rings, so the hash would already error out)

Fixes sagemath#35238

### 📝 Checklist

<!-- Put an `x` in all the boxes that apply. -->

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [x] I have updated the documentation and checked the documentation
preview.
    
URL: sagemath#40019
Reported by: nbruin
Reviewer(s): Martin Rubey

sagemathgh-40019: Hash fraction_field_elements more appropriately
    
<!-- ^ Please provide a concise and informative title. -->
<!-- ^ Don't put issue numbers in the title, do this in the PR
description below. -->
<!-- ^ For example, instead of "Fixes sagemath#12345" use "Introduce new method
to calculate 1 + 2". -->
<!-- v Describe your changes below in detail. -->
<!-- v Why is this change required? What problem does it solve? -->
<!-- v If this PR resolves an open issue, please link to it here. For
example, "Fixes sagemath#12345". -->

Equip various domains with a `canoncial_associate`, which finds unique
representatives in some rings modulo the action of the unit group. For
fields this is easy: there are only two orbits: 0 and the units (where 1
is a representative) and in ZZ we can just use "abs". For polynomial
rings we can extend representatives by normalizing the leading
coefficient to a canonical associate.

The normalization is important to find unique representatives of
fraction field elements (because multplying both numerator and
denominator by a unit doesn't change the represented class), which is
necessary for hashing them. The library uses hashes of fraction field
elements quite a bit, but luckily only in cases where we can actually
normalize denominators (in fact, "reduce" always seems to fail for such
rings, so the hash would already error out)

Fixes sagemath#35238

### 📝 Checklist

<!-- Put an `x` in all the boxes that apply. -->

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [x] I have updated the documentation and checked the documentation
preview.
    
URL: sagemath#40019
Reported by: nbruin
Reviewer(s): Martin Rubey