Scouring rush and field horsetail are two species in the Equisetum genus. These are primitive plants that produce spores rather than seeds, but they spread primarily by underground rhizomes. Until recently these two weeds were found primarily in roadside ditches, but the reduction in tillage has allowed them to spread into crop fields.
Although I suspect neither weed is highly competitive with corn or soybean, over time they can get dense enough to interfere with production. Few herbicides used in crop production have any effect on the equisetums. Tillage can suppress the weeds, but it probably would take several years of intensive tillage to eradicate them from fields.
Most infestations of these weeds are due to a source population in adjacent non-crop areas. Controlling the equisetum in these areas probably is the most efficient method of removing them from crop fields, but unfortunately this is not an easy task either. Repeated mowing or tillage is one option, but it will take at least two years of disturbance to control/eradicate the weed.
Chlorsuluron (Telar) is one herbicide that has good activity on equisetums, and it is registered for use in sites such as roadsides. Chlorsulfuron is a sulfonylurea herbicide (Group 2), related to products such as Accent and Classic. Multiple applications would be needed for complete control. It is a persistent compound that is toxic to both corn and soybean, so caution must be taken to prevent overspray into production fields
Chlorsulfuron is recommended at 1 to 2.6 oz/A (75% dry formulation) for controlling equisetums. An 8 oz container is the smallest quantity I could find on the internet. This container would provide sufficient herbicide to treat approximately a mile of a 20 ft roadside twice at an intermediate rate.
Counties vary in regulations regarding managing weeds in road rights of ways; determine local rules before implementing control strategies in these areas.
So how do you get rid of equisetum? The answer is: 'It ain't easy.' But then, managing a good weed never is.