Mom's Dill Pickles

My mother's dill pickles have always been extremely popular among not only our family, but anyone who came over to dinner. Mom always swore it was the well water from the farm that made them so good. Who knows -- maybe that's true. Here's her recipe, as best as could be divined, which yielded enough brine for about 3 to 4 pints. She indicated that the recipe could easily be doubled.

cucumbers, washed and sliced (Mom doesn't say how many; I used probably 3-4 medium-sized, plus a few cornichon sized, the latter of which are everyone's favorite)
4 cups water
2 cups white vinegar
6 tablespoons salt
1 teaspoon alum
1 garlic clove per jar (I don't remember us ever having fresh garlic on the farm, so this seems unusual coming from my mother, but I love garlic, so I included)
dill to taste (I used both fresh dill and dill seed; Mom probably used sprig of dill seed heads from her garden)
pinch of horseradish (I omitted this and can't remember my mother ever adding it, but she says she did)

In sterilized jars, place cucumbers, garlic cloves, dill and horseradish, if using. In a saucepan, combine water, vinegar, salt and alum. Bring to a boil and then pour over the cucumbers. Seal and process in a boiling water bath (I went for 10 minutes). Mom doesn't say, but I imagine one needs to let these rest at least a week or two before enjoying.