Who invented the horseshoe around 450 BC?

The horseshoe was invented by the Romans the poet Catullus mentioned mule that lost its horseshoe in a song from the first century BC.

It's unknown, for sure. Also, it depends on whether you're talking about the earliest ones, made of leather, or the more recent, made of metal.

http://inventors.about.com/od/famousinventions/fl/The-History-of-Horseshoes.htm

Because iron was a valuable commodity, and any worn out items were generally melted down and reused, it is difficult to locate clear archaeological evidence of the earliest horseshoes. From archaeological finds in Great Britain, it appears that the Romans attempted to protect their horses' feet with a strap-on, solid-bottomed "hipposandal" that has a slight resemblance to the modern hoof boot.

According to the Encyclopædia Britannica Eleventh Edition (1911),

Though the ancients were sufficiently impressed by the damage done to horses' hoofs to devise certain forms of covering for them (in the shape of socks or sandals), the practice of nailing iron plates or rim-shoes to the hoof does not appear to have been introduced earlier than the 2nd century B.C., and was not commonly known till the close of the 5th century A.D., or in regular use till the middle ages. The evidence for the earlier date depends on the doubtful interpretations of designs on coins, &c.
There is very little evidence of nailed-on shoes prior to AD 500 or 600, though there is speculation that the Gauls were the first to nail on metal horseshoes. The nailed iron horseshoe first appeared in the archaeological record in Europe about 5th century A.D. when a horseshoe, complete with nails, was found in the tomb of the Frankish King Childeric I at Tournai, Belgium. The earliest clear written record of iron horseshoes is a reference to "crescent figured irons and their nails" in AD 910.

https://en.wikipedia.org/wiki/Horseshoe#History

Since the early history of domestication of the horse, it was noted that working animals were exposed to many conditions that created breakage or excessive hoof wear. Ancient people recognized the need for the walls (and sometimes the sole) of domestic horses' hooves to have additional protection over and above any natural hardness. An early form of hoof protection was seen in ancient Asia where horse’s hooves were wrapped in rawhide, leather or other materials for both therapeutic purposes and protection from wear. The nailed shoe was a relatively late invention.

https://en.wikipedia.org/wiki/Horseshoe#History